Chain Rule Differentiation

Chain Rule Differentiation

When a function is built inside another function, use the chain rule. The idea is to differentiate the outside function first, then multiply by the derivative of the inside function.

1) Identify the layers

Look for an outer function and an inner function. For example, in ( (3x+1)^5 ), the outside is the power ((\cdot)^5) and the inside is (3x+1).

2) Differentiate the outer layer

Treat the inner expression as if it were a single variable. Then differentiate the outer function normally.

3) Multiply by the derivative of the inner layer

After differentiating the outside, multiply by the derivative of what is inside.

4) Simplify

Combine constants, reduce powers, and write the result in a clean form.

Quick check

Ask yourself: “Did I differentiate both the outside and the inside?” A common mistake is to stop after the outer derivative and forget the extra factor from the inner derivative.

Example pattern

If (y = f(g(x))), then [ \frac{dy}{dx} = f'(g(x)),g'(x). ] Use this repeatedly for nested expressions. After finding the derivative, simplify carefully to match the expected exact answer.