Chain Rule: Step-by-Step Guide
The chain rule differentiates a composition of functions: if y = f(g(x)), then dy/dx = f'(g(x)) * g'(x). In words: differentiate the outer function evaluated at the inner function, then multiply by the derivative of the inner function.
Math Racer skill guide · Calculus 1 · AP Calc AB / College
How to solve it, step by step
- Identify the outer and inner functions. Write the function as f(g(x)). The outer function f is the last operation applied; the inner function g is what sits inside it.\[y = f(g(x)), \;\; u = g(x)\]
- Differentiate the outer function with respect to u. Treat the inner function as a single variable u and find f'(u). Leave the inner function unchanged for now.\[\frac{df}{du} = f'(u)\]
- Differentiate the inner function. Compute g'(x), the derivative of the inside expression with respect to x.\[\frac{du}{dx} = g'(x)\]
- Multiply and substitute back. Multiply f'(u) by g'(x) and replace u with g(x) to express the derivative entirely in terms of x.\[\frac{dy}{dx} = f'(g(x)) \cdot g'(x)\]
Worked example
Find d/dx of (3x^2 + 1)^5.
Step 1: Identify the outer and inner functions. Outer: u^5; inner: u = 3x^2 + 1.
Step 2: Differentiate the outer with respect to u: d/du(u^5) = 5u^4.
Step 3: Differentiate the inner: d/dx(3x^2 + 1) = 6x.
Step 4: Apply the chain rule: dy/dx = 5u^4 * 6x = 30x * u^4.
Step 5: Substitute u = 3x^2 + 1 back: dy/dx = 30x(3x^2 + 1)^4.
Verification: At x = 1, original = 4^5 = 1024. Derivative = 30(1)(4)^4 = 30 * 256 = 7680. Numerical check: ((3*1.001^2+1)^5 - 4^5)/0.001 approximately 7693, close to 7680.
Tips and common mistakes
- Always identify the inner and outer functions before differentiating — write them down explicitly.
- The chain rule applies whenever you can describe the function as 'do this, then do that' — i.e., a composition.
- Common patterns: (something)^n, e^(something), ln(something), sin(something) — all need the chain rule.
- After the chain rule, immediately substitute the inner function back so your answer is purely in x.
- For nested compositions f(g(h(x))), apply the chain rule repeatedly: f'(g(h(x))) * g'(h(x)) * h'(x).
- Don't forget the multiplier — leaving off g'(x) is the most common chain-rule mistake.
- Pair the chain rule with product/quotient rules when the function is a composition multiplied or divided.
- Practice spotting chain rule first: if you see a function inside another function, the chain rule is required.