Derivatives

The Chain Rule

Differentiating composite functions using the chain rule.

Advanced20 min lesson2 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

Power, Product & Quotient Rules

Differentiating a Function Wrapped Inside Another

None of the previous rules directly handle (2x + 5)⁴ — it's one function wrapped around another. If y changes m times as fast as an in-between quantity u, and u changes n times as fast as x, y should change mn times as fast as x — rates compound by multiplication through a composition.

Definition — The Chain Rule

d/dx[f(g(x))] = f'(g(x)) · g'(x) — the outer function's derivative, evaluated at the inner function, times the inner function's own derivative.

\frac{d}{dx}\big[f(g(x))\big] = f'(g(x)) \cdot g'(x)

Worked Example — A Power of a Function

Differentiate y = (2x + 5)⁴. Outer f(u) = u⁴, f'(u) = 4u³; inner g(x) = 2x + 5, g'(x) = 2. dy/dx = 4(2x+5)³ · 2 = 8(2x+5)³.

Worked Example — A Trig Function of a Function

Differentiate y = sin(4x). Outer f(u) = sin(u), f'(u) = cos(u); inner g(x) = 4x, g'(x) = 4. dy/dx = cos(4x) · 4 = 4cos(4x). Check numerically near x = 0: dy/dx should be about 4cos(0) = 4; estimating directly, y(0) = 0 and y(0.001) = sin(0.004) ≈ 0.004, giving a slope of about 4 — matching.

Derivative Explorer

f(x) =

x^3 - 3*x

Point of tangency

x = a1

Show tangent line

Trace mode

f(x), with tangent line at x = a

Show f''(x)Show f'''(x)

Derivative graph — linked to the graph above; panning or zooming either moves both

What's happening at x = a

At x = 1, f(x) ≈ -2 — the point (1, -2).
The derivative there is f'(1) ≈ 0: the slope of the tangent line, and the instantaneous rate of change of f at this exact point.
That slope is positive, so f is increasing at this point.
The second derivative is positive here, so f is concave up (curving upward) near this point.
Near x = 1, the tangent line y ≈ -2 + 0·(x − 1) is f's best straight-line approximation — the core idea a derivative captures.

Numerical derivative comparison

Forward, backward, and central difference estimates of f'(a) as the step size h shrinks — all three should converge toward the same value.
h	Forward	Backward	Central
1	4	-2	1
0.1	0.31	-0.29	0.01
0.01	0.0301	-0.0299	0.0001
0.001	0.003	-0.003	0

Analysis of f(x)

y-intercept: (0, 0)
x-intercepts: (-1.73, 0), (0, 0), (1.73, 0)
Extrema: local max at (-1, 2); local min at (1, -2)
Inflection points: (0, 0)
Vertical asymptotes: none found in view
Horizontal asymptotes: none found
Domain: all real numbers in view
Range (estimated): approximately [-970, 970]

Analysis of f'(x) — its roots are f's critical points

y-intercept: (0, -3)
x-intercepts: (-1, 0), (1, 0)
Extrema: local min at (0, -3)
Inflection points: none found in view
Vertical asymptotes: none found in view
Horizontal asymptotes: none found
Domain: all real numbers in view
Range (estimated): approximately [-3, 297]

Tip

Ask 'what's the very last operation applied?' — that's the outer function; everything inside it is the inner function.

Common Mistakes

Differentiating (2x+5)⁴ as 4(2x+5)³ and stopping, without the inner derivative factor.
The chain rule always has two factors — dropping the inner derivative (here, ×2) is the most common chain rule error.

Key Takeaways

The chain rule differentiates f(g(x)) as f'(g(x)) · g'(x).
Identify the outer and inner functions by asking what operation applies last.

Summary

This closes the study of derivative rules. The next unit puts derivatives to work analyzing a function's own graph.

Track your progress

← Power, Product & Quotient Rules Back to Derivatives