Calculus · Quick Reference

Chain Rule Cheat Sheet

Differential Calculus · Lesson 5/20

In one line: the chain rule lets us differentiate composite functions — one function nested inside another — by working from the outside in.

Key Ideas

1Composite Functions. A function of the form f(g(x)), where g(x) is computed first and then plugged into f.

2The Chain Rule Formula. d/dx[f(g(x))] = f'(g(x)) * g'(x) — differentiate the outer function, keep the inner function unchanged, then multiply by the derivative of the inner function.

3Identifying Outer and Inner Functions. Ask 'what is being done last?' — that's usually the outer function. Everything inside it is the inner function.

4Chaining Multiple Layers. For deeply nested functions, apply the chain rule repeatedly, working outward to inward.

5Leibniz Notation View. If y = f(u) and u = g(x), then dy/dx = (dy/du) * (du/dx).

Worked Examples

Differentiate y = (3x + 1)^5.

dy/dx = 15(3x+1)^4

Differentiate y = sin(x^2).

dy/dx = 2x*cos(x^2)

Differentiate y = e^(4x).

dy/dx = 4e^(4x)

Formulas

d/dx[f(g(x))] = f'(g(x)) * g'(x)

dy/dx = (dy/du)*(du/dx)

Practice Yourself

Differentiate y = (x+5)^3.

3(x+5)^2

Differentiate y = cos(2x).

-2sin(2x)

Differentiate y = (2x^2+1)^4.

16x(2x^2+1)^3