## Composite Functions

A function *m* is a **composite** of functions *f* and *g* if

m(x) = f[g(x)]

The domain of *m* is the set of all numbers *x* such that *x* is in the domain of *g*, and *g(x)* is in the domain of *f*.

### General Power Rule

If *u(x)* is a differentiable function, *n* is any real number, and

y = f(x) = [u(x)]^{n}

then

f'(x) = n[u(x)]^{n-1}u'(x)

Using simplified notation,

y' = nu^{n-1}u'ord/dx u^{n} = nu^{n-1}du/dxwhereu = u(x)

### Chain Rule

If m(x) = E[I(x)] is a composite function, then

m'(x) = E'[I(x)]I'(x)

provided that E'[I(x)] and *I'(x)* exist. Equivalently, if y = E(u) and u = I(x), then

dy/dx = dy/du du/dx

provided that dy/du and du/dx exist.

### General Derivative Rules

- d/dx [f(x)]
^{n} = n[f(x)]^{n-1}f'(x)
- d/dx ln[f(x)] = f'(x)/[f(x)]
- d/dx e
^{f(x)} = e^{f(x)}f'(x)