Unit 3: Differentiation — Composite, Implicit & Inverse Functions

The Chain Rule

Differentiating composite functions using the chain rule.

Advanced25 min lesson2 min readUpdated August 12, 2026Author not yet attributed

Differentiating a Function Wrapped Inside Another

None of the rules so far can differentiate (4x − 1)³ directly — 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 = (4x − 1)³. Outer f(u) = u³, f'(u) = 3u²; inner g(x) = 4x − 1, g'(x) = 4. dy/dx = 3(4x−1)² · 4 = 12(4x−1)².

Worked Example — A Trig Function of a Function

Differentiate y = cos(3x²). Outer f(u) = cos(u), f'(u) = −sin(u); inner g(x) = 3x², g'(x) = 6x. dy/dx = −sin(3x²) · 6x = −6x sin(3x²). Check numerically at x = 1: the formula gives −6sin(3) ≈ −6(0.14112) ≈ −0.8467; estimating directly from y(1) = cos(3) ≈ −0.98999 and y(1.001) = cos(3.006003) ≈ −0.99084 gives a slope of about −0.847 — 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 (4x−1)³ as 3(4x−1)² and stopping, without the inner derivative factor.
The chain rule always has two factors — dropping the inner derivative (here, ×4) 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.
Rates compound multiplicatively through a composition — the intuition behind the rule's product of two derivatives.

Summary

The chain rule handles composite functions. The next lesson applies it to equations not solved for y, and to inverse functions.

Track your progress

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