Definition: The derivative of a function f at x , denoted by f'(x), is given by

f'(x) = lim_h->0 [f(x+h)-f(x)]/h
if this limit exists!

Corollary: The derivative of a constant is identically zero! This follows since f(x+h)=f(x) for all h...

The Power Rule: The derivative of f(x)=xⁿ, where n is a positive integer, is

d/dx f(x) = f'(x) = nx^n-1

Theorem:

1. (cf)' = cf'
2. (f+g)' = f' + g'
3. (f-g)' = f' - g'

Alas, we do not have (fg)' = f' g' ... We do, however, have

The Power Rule: d/dx f(x)g(x) = f(x) d/dx g(x) + g(x) d/dx f(x), or

(fg)' = f g' + g f'

If both derivative exist

The Quotient Rule: d/dx f(x)/g(x) = [g(x) d/dx f(x) - f(x) d/dx g(x)]/[g(x)²], or

(f/g)' = (g f' - f g')/g²

If both derivative exist , (and g does not vanish...)

Corollary: The derivative of f(x) = x^-n, where n is a positive integer, is given by

f'(x) = -n x ^-n-1

From the material in this section we now know how to differentiate in polynomial or rational function!

Last modified Wed Sep 18 20:59:23 CDT 1996