Calculus / Unit 4: Differentiation Rules
4.4 The Chain Rule
Notes
For a composition: $\displaystyle \frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$. Differentiate the outside, leave the inside alone, then multiply by the derivative of the inside.
- Example: $\displaystyle \frac{d}{dx}[(3x + 1)^5] = 5(3x + 1)^4 \cdot 3 = 15(3x + 1)^4$.
- For square roots: $\displaystyle \frac{d}{dx}\sqrt{u} = \frac{u'}{2\sqrt{u}}$.
- Forgetting the inside derivative is the most common mistake.
Practice
Find $\displaystyle \frac{d}{dx}[(3x + 1)^5]$.
Outside: $5u^4$. Inside derivative: 3. Multiply to get $15(3x + 1)^4$.
Find $\displaystyle \frac{d}{dx}\sqrt{x^2 + 1}$.
$\displaystyle \frac{2x}{2\sqrt{x^2 + 1}} = \frac{x}{\sqrt{x^2 + 1}}$.
Find $\displaystyle \frac{d}{dx}[(x^2 - 4x)^3]$.
Outside: $3u^2$. Inside derivative: $2x - 4$.
Ready for more? Take the Unit 4 test.