Differentiation J. Gerlach November 2010
<Text-field style="Heading 1" layout="Heading 1">D and diff</Text-field> The limit definition of the derivative is covered in the Limit tutorial. Here we look for direct ways to calculate derivatives. Maple has two commands for this task. One is the D operator, the other is the diff command. These commands behave quite differently, and we shall illustrate both for the example 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.
<Text-field style="Heading 2" layout="Heading 2"><Font size="14">The Use of D</Font></Text-field> D acts on functions, and we have to define f(x) first. 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 LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder Zio2I0kieEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKiQsJiIiIkYrKiQpRiQiIiNGK0YrISIiRiVGJUYl Now taking the derivative is straightforward, just enter 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 LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder LCQqKCIiIyIiIiksJkYlRiUqJClJInhHNiJGJEYlRiVGJCEiIkYqRiVGLA== If you type LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiREYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSShtZmVuY2VkR0YkNiQtRiM2JS1GLDYlUSJmRicvRjBRJXRydWVGJy9GM1EnaXRhbGljRicvJStleGVjdXRhYmxlR0YxRjJGMi1JI21vR0YkNi1RIjtGJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRj4vJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMjc3Nzc3OGVtRidGQUYy LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder Zio2I0kieEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCQqKCIiIyIiIiksJkYsRiwqJClGJEYrRixGLEYrISIiRiRGLEYxRiVGJUYl maple returns a function instead. This all goes back to distinction between a function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTEY5and a formula LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInhGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== for the output, the latter being an expression. In short, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiREYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJmRicvRjBRJXRydWVGJy9GM1EnaXRhbGljRidGMkYyRjI= stands for f', and therefore LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiREYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJmRicvRjBRJXRydWVGJy9GM1EnaXRhbGljRidGMkYyLUY2NiQtRiM2JC1GLDYlUSJ4RidGPUY/RjJGMi1JI21vR0YkNi1RIn5GJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGZm5GMg==represents f'(x). If you want to evaluate the derivative at a specified value, say you want the slope of our function at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjNGJ0Y5LUY2Ni1RIn5GJ0Y5RjtGPkZARkJGREZGRkgvRktRJjAuMGVtRicvRk5GV0Y5 you would enter 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 LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder IyEiJCIjXQ==
<Text-field style="Heading 2" layout="Heading 2"><Font size="14">The Use of diff</Font></Text-field> diff acts on expressions. You can enter the expression directly, and you do not have to define the function in advance. In our example we have the options 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 LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder LCQqKCIiIyIiIiksJkYlRiUqJClJInhHNiJGJEYlRiVGJCEiIkYqRiVGLA== or (the definition of f is still in effect) 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 LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder LCQqKCIiIyIiIiksJkYlRiUqJClJInhHNiJGJEYlRiVGJCEiIkYqRiVGLA== or in two steps LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSomY29sb25lcTtGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMjc3Nzc3OGVtRicvRk5GU0Y1LUkmbWZyYWNHRiQ2KC1JI21uR0YkNiRRIjFGJ0Y5LUYjNilGWC1GNjYtUSIrRidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjIyMjIyMjJlbUYnL0ZORlxvLUklbXN1cEdGJDYlLUYsNiVRInhGJ0YvRjItRiM2Jy1GWTYkUSIyRidGOUYvLyUrZXhlY3V0YWJsZUdGPS8lMGZvbnRfc3R5bGVfbmFtZUdRKDJEfk1hdGhGJ0YyLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0YvRmlvRltwRjIvJS5saW5ldGhpY2tuZXNzR0Zlbi8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZlcC8lKWJldmVsbGVkR0Y9LUY2Ni1RIjtGJ0Y5RjsvRj9GMUZARkJGREZGRkhGSkZURmlvRjk= LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder KiQsJiIiIkYkKiQpSSJ4RzYiIiIjRiRGJCEiIg== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVElZGlmZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYnLUYsNiVRInlGJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRInhGJ0YvRjIvJStleGVjdXRhYmxlR0ZFRkFGQS1GPjYtUSI7RidGQUZDRkZGSEZKRkxGTkZQRlIvRlZRLDAuMjc3Nzc3OGVtRidGZW5GQQ== LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder LCQqKCIiIyIiIiksJkYlRiUqJClJInhHNiJGJEYlRiVGJCEiIkYqRiVGLA== In all cases the command has the form diff( expression, variable). The template on the left executes the diff command 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 LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder LCQqKCIiIyIiIiksJkYlRiUqJClJInhHNiJGJEYlRiVGJCEiIkYqRiVGLA== Evaluation of the derivative at a point requires two steps (use LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjNGJ0Y5LUY2Ni1RIn5GJ0Y5RjtGPkZARkJGREZGRkgvRktRJjAuMGVtRicvRk5GV0Y5again). 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 LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder LCQqKCIiIyIiIiksJkYlRiUqJClJInhHNiJGJEYlRiVGJCEiIkYqRiVGLA== 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 LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder IyEiJCIjXQ== JSFH
<Text-field style="Heading 2" layout="Heading 2"><Font size="14">Do not mix D and diff</Font></Text-field> The following results in nonsense (the definitions of f(x) and y are still in effect). 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 LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder LCQqKiIiIyIiIi0tSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJInhHRixGLUYlLUYuRi1GJSksJkYlRiUqJClGL0YkRiVGJUYkISIiRjQ= 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 LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder IiIh
<Text-field style="Heading 2" layout="Heading 2"><Font size="14">Clickable Math</Font></Text-field> Enter the expression and right-click. 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 LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder KiQsJiIiIkYkKiQpSSJ4RzYiIiIjRiRGJCEiIg== LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi5RJyZyYXJyO0YnLyUwZm9udF9zdHlsZV9uYW1lR1EoMkR+TWF0aEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R1EldHJ1ZUYnLyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGNy8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZHLUkmbXRleHRHRiQ2JVE3ZGlmZmVyZW50aWF0ZX53LnIudC5+eEYnRi9GMkZD 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 LCQqKCIiIyIiIiksJkYlRiUqJClJInhHNiJGJEYlRiVGJCEiIkYqRiVGLA== LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi5RJyZyYXJyO0YnLyUwZm9udF9zdHlsZV9uYW1lR1EoMkR+TWF0aEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R1EldHJ1ZUYnLyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGNy8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZHLUkmbXRleHRHRiQ2JVEyZXZhbHVhdGV+YXR+cG9pbnRGJ0YvRjJGQw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY1LUkjbWlHRiQ2KVElZXZhbEYnLyUnZmFtaWx5R1EwVGltZXN+TmV3flJvbWFuRicvJSVib2xkR1EmZmFsc2VGJy8lJ2l0YWxpY0dRJXRydWVGJy8lK2ZvcmVncm91bmRHUShbMCwwLDBdRicvJStleGVjdXRhYmxlR0Y0LyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjFRIihGJ0YvRjJGOEY7L0Y+USdub3JtYWxGJy8lJmZlbmNlR0Y3LyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjcvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMTY2NjY2N2VtRicvJSdyc3BhY2VHRlYtRkE2MVEifkYnRi9GMkY4RjtGRC9GR0Y0RkgvRktGNEZMRk5GUEZSL0ZVUSYwLjBlbUYnL0ZYRmluLUkobWFjdGlvbkdGJDYlLUkqbXZlcmJhdGltR0YkNiNRTCwkKigiIiMiIiIpLCZGJUYlKiQpSSJ4RzYiRiRGJUYlRiQhIiJGKkYlRixGJy8lK2FjdGlvbnR5cGVHUTltYXBsZXNvZnQuY29tOmxhYmVsKEw4NilGJy8lJXZpZXdHUSZsYWJlbEYnLUZBNjFRIixGJ0YvRjJGOEY7RkRGZm4vRklGN0ZnbkZMRk5GUEZSRmhuL0ZYUSwwLjMzMzMzMzNlbUYnRlktRkE2MVEiW0YnRi9GMkY4RjtGREZGRkhGSkZMRk5GUEZSRlRGVy1GLDYpUSJ4RidGL0YyRjVGOEY7Rj1GWS1GQTYxUSI9RidGL0YyRjhGO0ZERmZuRkhGZ25GTEZORlBGUi9GVVEsMC4yNzc3Nzc4ZW1GJy9GWEZocEZZLUkjbW5HRiQ2KFEiM0YnRi9GMkY4RjtGRC1GQTYxUSJdRidGL0YyRjhGO0ZERkZGSEZKRkxGTkZQRlJGVC9GWFEsMC4xMTExMTExZW1GJy1GQTYxUSIpRidGL0YyRjhGO0ZERkZGSEZKRkxGTkZQRlJGVEZXRi9GMkY4RjtGRA== IyEiJCIjXQ==
<Text-field style="Heading 2" layout="Heading 2"><Font size="14">More Examples</Font></Text-field> Problem: Find the derivative of 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. 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 LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder Zio2I0kieEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKiYpRiQiIiMiIiItSSRleHBHRiU2IywkRiQhIiJGLEYlRiVGJQ== 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 LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder LCYqKCIiIyIiIkkieEc2IkYlLUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YnNiMsJEYmISIiRiVGJSomKUYmRiRGJUYoRiVGLw== Problem: Find the derivative of 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 and simplify the result 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 LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder Zio2I0kieEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKiYpLCYqJiIiIyIiIkYkRi5GLkYuRi4iIipGLiksJiomIiImRi5GJEYuRi5GLSEiIiIjN0YuRiVGJUYl 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 LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder LCYqKCIjPSIiIiksJiomIiIjRiVJInhHNiJGJUYlRiVGJSIiKUYlKSwmKiYiIiZGJUYqRiVGJUYpISIiIiM3RiVGJSooIiNnRiUpRiciIipGJSlGLiIjNkYlRiU= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEnZmFjdG9yRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNiUtRiw2JVEiJUYnRi9GMi8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJ0ZALUkjbW9HRiQ2LVEiO0YnRkAvJSZmZW5jZUdGPy8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0Y/LyUqc3ltbWV0cmljR0Y/LyUobGFyZ2VvcEdGPy8lLm1vdmFibGVsaW1pdHNHRj8vJSdhY2NlbnRHRj8vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4yNzc3Nzc4ZW1GJ0Y9RkA= LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder LCQqKiIiJyIiIiwmIiIlRiUqJiIjTkYlSSJ4RzYiRiVGJUYlKSwmKiYiIiNGJUYqRiVGJUYlRiUiIilGJSksJiomIiImRiVGKkYlRiVGLyEiIiIjNkYlRiU= Problem: Find the slope of the function 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 at the point where 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 . 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 LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder Zio2I0kieEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKiQpLUkkc2luRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlRiMiIiMiIiJGJUYlRiU= 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 LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder LCQqJiMiIiIiIiNGJSkiIiRGJEYlISIi
<Text-field style="Heading 1" layout="Heading 1">Higher Order Derivatives</Text-field>
<Text-field style="Heading 2" layout="Heading 2"><Font size="14">D and diff</Font></Text-field> Let's say we want to compute the 5th derivative of the function 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. Using the D operator, we define the function and then use a modified D command, namely (D@@n), where n is the order of the derivative. 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 LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder Zio2I0kieEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKiYpRiQiIiMiIiItSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IywkRiQhIiJGLEYlRiVGJQ== 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 LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder LCgqJiIjPyIiIi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCRJInhHRishIiJGJUYvKigiIzVGJUYuRiVGJkYlRiUqJilGLiIiI0YlRiZGJUYv There appears to be a common factorLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEnZmFjdG9yRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNiUtRiw2JVEiJUYnRi9GMi8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJ0ZALUkjbW9HRiQ2LVEiO0YnRkAvJSZmZW5jZUdGPy8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0Y/LyUqc3ltbWV0cmljR0Y/LyUobGFyZ2VvcEdGPy8lLm1vdmFibGVsaW1pdHNHRj8vJSdhY2NlbnRHRj8vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4yNzc3Nzc4ZW1GJ0Y9RkA= LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder LCQqJi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCRJInhHRikhIiIiIiIsKCIjP0YuKiYiIzVGLkYsRi5GLSokKUYsIiIjRi5GLkYuRi0= With the diff command, we repeatedly enter x, once of each derivative. 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 LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder LCgqJiIjPyIiIi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCRJInhHRishIiJGJUYvKigiIzVGJUYuRiVGJkYlRiUqJilGLiIiI0YlRiZGJUYv This can get rather lengthy if you need a real high order derivative. Fortunately, there is a fix for this problem using x$n. 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 LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder LCgqJiIjPyIiIi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCRJInhHRishIiJGJUYvKigiIzVGJUYuRiVGJkYlRiUqJilGLiIiI0YlRiZGJUYv
<Text-field style="Heading 2" layout="Heading 2"><Font size="14">Higher Order Derivatives at a Point</Font></Text-field> Evaluation of a higher order derivative at a specified point is again a little simpler with the D operator. Let's compute the eight derivative of 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 and evaluate it at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NjA7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy. First we will use the D operator. 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 LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder Zio2I0kieEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKiZGJCIiIi1JJHNpbkdGJUYjRipGJUYlRiU= 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 LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder IiIp With the diff command we proceed as follows. 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 LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder LCYqJiIiKSIiIi1JJGNvc0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSJ4R0YrRiUhIiIqJkYtRiUtSSRzaW5HRihGLEYlRiU= 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 LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder LCYqJiIiKSIiIi1JJGNvc0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSNQaUdGKUYlISIiKiZGLUYlLUkkc2luR0YoRixGJUYl LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEmdmFsdWVGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2JS1GLDYlUSIlRidGL0YyLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjNRJ25vcm1hbEYnRkAtSSNtb0dGJDYtUSI7RidGQC8lJmZlbmNlR0Y/LyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRj8vJSpzeW1tZXRyaWNHRj8vJShsYXJnZW9wR0Y/LyUubW92YWJsZWxpbWl0c0dGPy8lJ2FjY2VudEdGPy8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjI3Nzc3NzhlbUYnRj1GQA== LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder IiIp
<Text-field style="Heading 2" layout="Heading 2"><Font size="14">More Examples</Font></Text-field> Problem: Compute f'(x) and f''(x) when 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. 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 LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder LCYqKCIiIyIiIkkieEc2IkYlLCYtSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRic2I0YmRiVGJUYlRiVGJSomLCYqJClGJkYkRiVGJUYlRiVGJUYpRiVGJQ== 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 LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder LCoqKCIiIyIiIkkieEc2IkYlLUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YnNiNGJkYlRiUqJkYkRiVGJkYlRiUqJkYoRiUpRiZGJEYlRiVGKEYl 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 LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder LCoqJiIiJCIiIi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSJ4R0YrRiVGJSIiI0YlKigiIiVGJUYtRiVGJkYlRiUqJkYmRiUpRi1GLkYlRiU= or 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 LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder Zio2I0kieEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKiYsJiokKUYkIiIjIiIiRi5GLkYuRi4sJi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJUYjRi5GLkYuRi5GJUYlRiU= 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 LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder LCoqKCIiIyIiIkkieEc2IkYlLUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YnNiNGJkYlRiUqJkYkRiVGJkYlRiUqJkYoRiUpRiZGJEYlRiVGKEYl 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 LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder LCoqJiIiJCIiIi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSJ4R0YrRiVGJSIiI0YlKigiIiVGJUYtRiVGJkYlRiUqJkYmRiUpRi1GLkYlRiU= Problem: Compute the 15th derivative of 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. 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 LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder LCQqJiImb0YkIiIiLUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMsJComIiIjRiVJInhHRitGJSEiIkYlRjE=
<Text-field style="Heading 1" layout="Heading 1">More Examples</Text-field>
<Text-field style="Heading 2" layout="Heading 2"><Font size="14">A Function and its Derivative in a Common Graph</Font></Text-field> Problem: Sketch the function 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and its derivative in a common figure. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY0LUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSomY29sb25lcTtGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMjc3Nzc3OGVtRicvRk5GU0Y1LUYsNiVRInhGJ0YvRjJGNS1GNjYtUSgmc3JhcnI7RidGOUY7Rj5GQEZCRkRGRkZIRkpGTUY1LUYsNiVRJHNpbkYnL0YwRj1GOS1JKG1mZW5jZWRHRiQ2JC1GIzYoRlUtRjY2LVEoJm1pbnVzO0YnRjlGO0Y+RkBGQkZERkZGSC9GS1EsMC4yMjIyMjIyZW1GJy9GTkZib0Zlbi1Gam42JC1GIzYlRlUvJStleGVjdXRhYmxlR0Y9RjlGOUZob0Y5RjktRjY2LVEiO0YnRjlGOy9GP0YxRkBGQkZERkZGSEZKRlRGNUY1RjUtRjY2LVEiI0YnRjlGO0Y+RkBGQkZERkZGSEZKRk0tRiw2JVE8fkRlZmluaXRpb25+b2Z+dGhlfmZ1bmN0aW9uRidGL0YyRmhvRjk= LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder Zio2I0kieEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUkkc2luRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiMsJkYkIiIiLUYqRiMhIiJGJUYlRiU= 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The graph of the function is in red, its derivative in blue. It is evident that the blue curve measures the slope of the red curve.
<Text-field style="Heading 2" layout="Heading 2"><Font size="14">Function with a Tangent Line in a Common Figure</Font></Text-field> Problem: Graph the function 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 and its tangent line at x=4 in a common figure. 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 LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder 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 LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder LCYiIiJGIyomI0YjIiIlRiNJInhHNiJGI0Yj 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<Text-field style="Heading 2" layout="Heading 2"><Font size="14">Find Horizontal Tangent Lines</Font></Text-field> Problem: Find all points in the interval 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 for which the graph of 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 has a horizontal tangent line. In order to solve the problem, we need to calculate the derivative , set it to zero, and solve the resulting equation for x. 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Let's look at the graph of f. 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Let's see what we get from fsolve, and label the points. 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 LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder JCIrZzkpKTRCISIq 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 LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder JCIrWlFJdFIhIio= 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 LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder JCIiIUYj 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 LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder JCIrMWA9JEcnISIq Here are the coordinates of the points: 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 LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder NiQkIiIhRiRGIw== NiQkIitnOSkpNEIhIiokIiIiIiIh NiQkIitaUUl0UiEiKiQhIiIiIiE= NiQkIisxYD0kRychIiokIStxWidleiIhIz4= JSFH
<Text-field style="Heading 1" layout="Heading 1">Implicit Differentiation</Text-field>
<Text-field style="Heading 2" layout="Heading 2"><Font size="14">The implicitdiff Command</Font></Text-field> Here we take the derivative on both sides of an equation. We need to set up the equation, and decide on the dependent and the independent variable. We will use 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 as an example. Note that the point (1,2) lies on this curve, and we shall determine the slope there. First we define the equation for future reference. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdRJjAuMGVtRicvJSZkZXB0aEdGNC8lKmxpbmVicmVha0dRKG5ld2xpbmVGJ0YrLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJStleGVjdXRhYmxlR0Y9Rjk= print(); # input placeholder 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 LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder LywmKiQpSSJ4RzYiIiIkIiIiRikqJClJInlHRidGKEYpRiksJCooIyIiKiIiI0YpRiZGKUYsRilGKQ== First we confirm that the point (1,2) lies on the curve. To do so, we substitute the coordinates into the equation. 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 LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder LyIiKkYj It worked!. Now we use the command implicitdiff. It has the structure implicitdiff( equation, dependent variable, independent variable). In our case we have 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 LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder KiYsJiomIiIjIiIiKUkieEc2IkYlRiYhIiIqJiIiJEYmSSJ5R0YpRiZGJkYmLCYqJkYlRiYpRi1GJUYmRiYqJkYsRiZGKEYmRipGKg== This is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbWZyYWNHRiQ2KC1JI21pR0YkNiVRI2R5RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiUtRi82JVEjZHhGJ0YyRjVGMkY1LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZCLyUpYmV2ZWxsZWRHUSZmYWxzZUYnL0Y2USdub3JtYWxGJw==, representing the slope. If we substitute the coordinates of our point, we obtain 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 LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder IyIiJSIiJg== Let's illustrate the result graphically. In order to graph the equation, we need a more sophisticated plotting command, and we have to load the plots package. Now the command implicitplot will do the desired task. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEld2l0aEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRJnBsb3RzRidGL0YyLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjNRJ25vcm1hbEYnRkAtSSNtb0dGJDYtUSI6RidGQC8lJmZlbmNlR0Y/LyUqc2VwYXJhdG9yR0Y/LyUpc3RyZXRjaHlHRj8vJSpzeW1tZXRyaWNHRj8vJShsYXJnZW9wR0Y/LyUubW92YWJsZWxpbWl0c0dGPy8lJ2FjY2VudEdGPy8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlZGPUZA LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEtaW1wbGljaXRwbG90RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNjUtSSNtb0dGJDYtUSJ+RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtRiw2JVEkRXFuRidGL0YyLUY7Ni1RIixGJ0Y+RkAvRkRGMUZFRkdGSUZLRk1GTy9GU1EsMC4zMzMzMzMzZW1GJ0Y6LUYsNiVRInhGJ0YvRjItRjs2LVEiPUYnRj5GQEZDRkVGR0ZJRktGTS9GUFEsMC4yNzc3Nzc4ZW1GJy9GU0Zeby1GOzYtUSomdW1pbnVzMDtGJ0Y+RkBGQ0ZFRkdGSUZLRk0vRlBRLDAuMjIyMjIyMmVtRicvRlNGZG8tSSNtbkdGJDYkUSIxRidGPi1GOzYtUSMuLkYnRj5GQEZDRkVGR0ZJRktGTUZjb0ZSLUZnbzYkUSIzRidGPkZXRjotRiw2JVEieUYnRi9GMkZqbkZgb0Zmb0Zqb0ZdcEY+Rj4tRjs2LVEiO0YnRj5GQEZaRkVGR0ZJRktGTUZPRl9vLyUrZXhlY3V0YWJsZUdGQkY+ LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= %; 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 Now let's include the tangent line as well (I used the point-slope form without solving for y first). 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<Text-field style="Heading 2" layout="Heading 2"><Font size="14">More Examples</Font></Text-field> Problem: Find dy/dx from the equation y = sin(xy). 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 LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder LCQqKC1JJGNvc0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjKiZJInhHRikiIiJJInlHRilGLUYtRi5GLSwmKiZGJEYtRixGLUYtRi0hIiJGMUYx Problem: What is the slope of the curve given by 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 at the point (9,4)? 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 LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder LywmKiQpSSJ4RzYiIyIiIiIiI0YpRikqJClJInlHRidGKEYpRikiIiY= 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 LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder LCQqJilJInlHNiIjIiIiIiIjRigpSSJ4R0YmRichIiJGLA== 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 LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder LCQqKCMiIiIiIipGJSkiIiUjRiUiIiNGJSlGJkYpRiUhIiI= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEpc2ltcGxpZnlGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2JS1GLDYlUSIlRidGL0YyLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjNRJ25vcm1hbEYnRkAtSSNtb0dGJDYtUSI7RidGQC8lJmZlbmNlR0Y/LyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRj8vJSpzeW1tZXRyaWNHRj8vJShsYXJnZW9wR0Y/LyUubW92YWJsZWxpbWl0c0dGPy8lJ2FjY2VudEdGPy8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjI3Nzc3NzhlbUYnRj1GQA== LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder IyEiIyIiJA== Problem: Find the points with horizontal tangents on the curve 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. First define the curve as C QyQ+SSJDRzYiLyokLCYqJEkieEdGJSIiIyIiIiokSSJ5R0YlRitGLEYrLCQqJkYqRitGLkYsIiIlRiw= LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder LyokLCYqJEkieEc2IiIiIyIiIiokSSJ5R0YnRihGKUYoLCQqJkYmRihGK0YpIiIl Derivative by implicit differentiation QyQ+SSNkQ0c2Ii1JLWltcGxpY2l0ZGlmZkdGJTYlSSJDR0YlSSJ5R0YlSSJ4R0YlIiIi LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder LCQqKEkieEc2IiIiIiwoKiRGJCIiI0YmKiRJInlHRiVGKUYmRishIiNGJiwoKiZGJEYpRitGJkYmKiRGKyIiJEYmRighIiJGMUYx Now set the derivative to zero. QyQtSSZzb2x2ZUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkL0kjZENHRigiIiFJInhHRigiIiI= LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder NiUiIiEqJCwmKiRJInlHNiIiIiMhIiJGJ0YpIyIiIkYpLCRGJEYq Clearly, one solution occurs when x=y=0. The other two occur when 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 or the negative of it. In order to identify the points, we need to substitute these values into the original equation. As an alternative, we could solve the original equation along with dC=0 simultaneously. QyQtSSZzb2x2ZUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkPCRJIkNHRigvSSNkQ0dGKCIiITwkSSJ4R0YoSSJ5R0YoIiIi LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder NiQ8JC9JInhHNiIhIiIvSSJ5R0YmIiIiPCQvRiVGKkYo Now we have the solutions (-1,1) and (1,1). Maple probably ignored the origin, because the derivative is undefined at this point.
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