Econ 407: Lecture notes on derivatives. Economists look at things at the margin; that is they want to know what is the effect of a small ( marginal) change in a single variable (holding all other variables constant) on some variable of interest. To this end, they use such tools as the "derivative" or "partial derivative". A derivative simply measures the rate of change in one variable as another variable is changed by a small amount. For example, in the following demand function, we look to know what happens to quantity demanded if the price changes by a small amount. For a linear demand curve or any linear function, the derivative is going to be a constant, meaning that the rate of change does not change as move from one point to another on the demand schedule. For a polynomial, the derivative will be a function, assuming different values at different points.

qd:=100-5*p;

> d(qd)/d(p)=-5;

> diff(qd,p);

> qd:=100-5*p^2;

> dqd:=diff(qd,p);

> plot({qd,dqd},p=0..5);

> y:=x^2+6*x-7;

> dy:=diff(y,x);

>

> plot({y,dy},x=-10..8);

>

> solve(dy,x);

> eval(dy,x=-3);

> eval(y,x=-3);

> y:=8*x^3;

> dy:=diff(y,x);

> y:=(10*x^8-6*x^7)/(2*x);

> dy:=diff(y,x);

> simplify(dy);

>

> y:=(4*x^5.0)/(1-3*x);

> dy:=diff(y,x);

> simplify(dy);

Note that in all the instances where the exponent of x is greater than one, the derivative is a function of x.

> y:= 1/(x);

y := 1/x

> dy:=diff(y,x);

dy := -1/x^2

> plot(y,x= -01..1);

> plot(dy,x=-0.5..0.5);

>