M171 Calculus I · Unit 2: Computing derivatives · on Exam 2
§2.5 The chain rule
Study notes for The chain rule in M171 Calculus I at Montana State University: the big ideas, key formulas, a worked example, common mistakes and an exam tip. The same topic has endless practice questions and flashcards in StudyCat.
Big ideas
- For a composite $f(g(x))$: differentiate the outside function, evaluate it at the inside, then multiply by the derivative of the inside.
- Spot the inside function: anything "wrapped" in a power, root, exponential, log or trig function. $e^{3x^2}$ has inside $3x^2$; $(x^2+1)^5$ has inside $x^2+1$; $\sin(\cos x)$ has inside $\cos x$.
- Chain rule combines with every other rule: $\frac{d}{dx}[x^2 e^{5x}] = 2xe^{5x} + 5x^2e^{5x}$ (product rule with a chain inside).
Key formulas
Chain rule
$$\frac{d}{dx}\big[f(g(x))\big] = f'(g(x))\,g'(x)$$
Leibniz form
$$\frac{dy}{dx} = \frac{dy}{du}\cdot\frac{du}{dx}$$
Common shapes
$$\frac{d}{dx}[e^{kx}] = ke^{kx},\quad \frac{d}{dx}[\sin(kx)] = k\cos(kx),\quad \frac{d}{dx}[(u)^n] = n u^{n-1}u'$$
Worked example
Differentiate $g(x) = \sin(3x^2)$ and $h(x) = (2x^3 - 5)^4$.
Show the solution
$g'(x) = \cos(3x^2)\cdot 6x = 6x\cos(3x^2)$. $h'(x) = 4(2x^3-5)^3\cdot 6x^2 = 24x^2(2x^3-5)^3$.
Common mistakes
- Forgetting to multiply by the inside derivative (the most common calculus error).
- Differentiating the inside "in place": $\frac{d}{dx}[\sin(3x^2)] \ne \sin(6x)$.
Exam tipSay it out loud as you write: "derivative of the outside, inside unchanged, times derivative of the inside."
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