3.1 The Chain Rule

What the Chain Rule does

The Chain Rule lets you differentiate composite functions — functions inside other functions. If h(x) = f(g(x)), then h'(x) = f'(g(x)) · g'(x). You take the derivative of the outer function, keep the inner function unchanged inside it, then multiply by the derivative of the inner function.

Layers and Leibniz notation

In Leibniz notation, if y = f(u) and u = g(x), then dy/dx = (dy/du)(du/dx). For functions with multiple layers, apply the rule repeatedly, working from outside in. Each layer contributes its own derivative factor — miss one and the answer is wrong.

Common patterns

Watch for power of a function (e.g., (3x²+1)⁵), trig of a function (sin(x³)), exponential of a function (e^(2x)), and nested combos. Always identify the outermost operation first, differentiate it, then multiply by the derivative of what's inside — repeating as needed.

Key terms

Chain Rule formula

If h(x) = f(g(x)), then h'(x) = f'(g(x)) · g'(x).

Chain Rule in Leibniz notation

dy/dx = (dy/du)(du/dx), where y depends on u and u depends on x.

Steps to apply the Chain Rule

1) Identify outer and inner functions. 2) Differentiate the outer function, keeping the inner function unchanged. 3) Multiply by the derivative of the inner function.

Derivative of e^(g(x))

e^(g(x)) · g'(x)

Derivative of sin(g(x))

cos(g(x)) · g'(x)

Try an AP-style question

d/dx sin(x²) = ?

1. cos(2x)
2. 2x cos(x²)
3. cos(x²)
4. 2x sin(x²)