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MediumDerivativesCalculus0:49AP Calculus AB, unit 3

The Chain Rule

Differentiate (3x² + 1)⁴ with the chain rule: derivative of the outside, times derivative of the inside, giving 24x(3x² + 1)³.

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Every board from the lesson, so you can read it at your own pace or copy it into your notes.

1. Spot the composition0:00
1. $y = (3x^2 + 1)^4$
2. $\text{outside: } u^4 \qquad \text{inside: } u = 3x^2 + 1$
2. The rule0:14
$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$
outside derivative times inside derivative
3. Apply it0:23
1. $\frac{dy}{du} = 4u^3 = 4(3x^2 + 1)^3$
2. $\frac{du}{dx} = 6x$
3. $\frac{dy}{dx} = 4(3x^2 + 1)^3 \cdot 6x = 24x(3x^2 + 1)^3$
4. Recap0:39
- Name the inside function $u$ .
- Differentiate the outside, leave $u$ alone.
- Multiply by $u'$ . Forgetting it is the most common mistake.

About derivatives

Power, product, quotient and chain rules.

Practise what you just watched

\frac{d}{dx}\left(x^3\right)

A direct application of the power rule.

Find the derivative of

x^2 \sin x

A product of two functions needs the product rule.

e^{3x}

The chain rule with a linear inner function.

\ln(x^2 + 1)

Chain rule with the natural logarithm.

Derivative Calculator: try the same kind of problem with your own numbers.
Implicit Differentiation Calculator: try the same kind of problem with your own numbers.
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Limits Study Guide (study guide)
AP Calculus AB Unit 3: Differentiation: Composite, Implicit and Inverse Functions: the AP unit this lesson supports.
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