The Chain Rule

Quite possibly one of the most useful rules.

The Chain Rule. To differentiate the composite function f(x) = g(u(x)), first find the deriative of g as a function of u, then multiply the result by the derivative of u. It's much like a function of a function.
  The derivative of y = mx+b is m, the slope of the line y.
  The derivative of a term x raised to a power n is just as shown.
   

Let's show an example to hopefully make it clearer.
f(x) = sqrt(x^3-x^2)
Let u(x) = x^3-x^2, then u'(x) = 3x^2-2x. u(x) is the inside function in this case.
f(x) can be rewritten as f(x) = (x^3-x^2)^1/2.
Thus, g(u) = u^1/2.
Apply the power rule here..
g'(u) = 1/2 * u^(-1/2).
Okay, now back to the chain rule, substitute things in and multiply it out.
f'(x) = 1/2 * (x^3-x^2)^(-1/2) * (3x^2-2x)
f'(x) = 1/2 * 1/(x^3-x^2)^12 * (3x^2-2x)
f'(x) = 3/2*x^2-x

that was an ugly example.

Here are some more for practice.
1. m(x) = (3x^2+x)^4, find m'(x).

2. a(x) = 1/(x^5-2)^5, find a'(x).

3. t(x) = sqrt((x^5+3)/x^2), find t'(x).

4. h(x) = (x^2+3)^3, find h'(x).