MAT200
Franklin College
Erich Prisner

Laws of Differentiation

The laws of differentiation all rely on the limit laws that have been proven in the Limit Game page.

Power Rule

To derive the general power rule, one needs the binomial theorem, which is fairly easy to derive for natural exponent  r, but a little more difficult for general  r.
Still another possibility is to show it using logarithmic differentiation: Let f(x)=x^r. Then ln(f(x)=rln(x) has a derivative of r/x, hence f'(x)=f(x)*r/x = rx^r-1. (Note that logarithmic differentiation relies on chain rule and differentiating the logarithmic function, both topics usually treated later than the power rule.)

Sum Rule

Product Rule

Go on and finish the proof. At each step, explain which limit law you used.

Chain Rule

Try to give a proof along these lines as well.

Exponential/Logarithmic Functions

Proving these is more difficult and requires a theorem on e as a limit.

Erich Prisner, October 2003