Section 3.4: The Chain Rule

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-1u'(x)

Using simplified notation,

y' = nu^n-1u'ord/dx uⁿ = nu^n-1du/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

1. d/dx [f(x)]ⁿ = n[f(x)]^n-1f'(x)
2. d/dx ln[f(x)] = f'(x)/[f(x)]
3. d/dx e^f(x) = e^f(x)f'(x)