Math · Calculus I · Worked example

Differentiate a function with three layers

Differentiate y = sin³(4x).

y = {sin}^{3} (4 x)

Read the notation

sin³(4x) means (sin 4x)³. The cube applies to the whole sine, not to the angle 4x.

y = {(sin (4 x))}^{3}

Count the layers

Work it out for x = 1: first 4x = 4, then sin 4, then the cube of that. Three operations mean three layers: 4x is the innermost, the sine is the middle and the cube is the outermost. Give each inner layer its own letter.

v = 4 x, u = sin v, y = u^{3}

Differentiate each layer

Differentiate each layer with respect to the layer just inside it.

\frac{d y}{d u} = 3 u^{2}, \frac{d u}{d v} = cos v, \frac{d v}{d x} = 4

\begin{array}{c} \frac{d y}{d u} = 3 u^{2} \\ \frac{d u}{d v} = cos v \\ \frac{d v}{d x} = 4 \end{array}

Multiply all three rates

A chain of three layers multiplies three rates.

\frac{d y}{d x} = \frac{d y}{d u} \cdot \frac{d u}{d v} \cdot \frac{d v}{d x} = 3 u^{2} \cdot cos v \cdot 4

\begin{aligned} \frac{d y}{d x} & = \frac{d y}{d u} \cdot \frac{d u}{d v} \cdot \frac{d v}{d x} \\ = 3 u^{2} \cdot cos v \cdot 4 \end{aligned}

Put the layers back

Work back toward x: replace u with sin(4x), and v with 4x.

\frac{d y}{d x} = 3 {sin}^{2} (4 x) \cdot cos (4 x) \cdot 4

Simplify

Multiply the constants: 3·4 = 12.

\frac{d y}{d x} = 12 {sin}^{2} (4 x) cos (4 x)

The fast way

Once you can see the layers, skip the letters and write one factor per layer, from the outside in, copying everything inside each layer unchanged: 3 sin²(4x) for the cube, cos(4x) for the sine and 4 for 4x.

Result

dy/dx = 12 sin²(4x) cos(4x).

\frac{d y}{d x} = 12 {sin}^{2} (4 x) cos (4 x)

Your turn

Differentiate y = cos²(3x).

Show the answer and explanation

dy/dx = −6 cos(3x) sin(3x).

cos²(3x) means (cos 3x)², which has three layers: the square, the cosine and 3x. The square gives 2 cos(3x), the cosine gives −sin(3x) and 3x gives 3. Multiply: 2 cos(3x)·(−sin(3x))·3 = −6 cos(3x) sin(3x). With the double-angle identity this is also −3 sin(6x).

\begin{array}{l} y = {cos}^{2} (3 x) \\ \frac{d y}{d x} = 2 cos (3 x) \cdot (- sin (3 x)) \cdot 3 \\ \frac{d y}{d x} = - 6 cos (3 x) sin (3 x) \end{array}

Keep exploring

Open the steps in Math: the Checker marks the derivative ✓. Remove the factor 4 and its mark turns to ✗, because the innermost layer’s factor is missing.

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OpenStax Calculus Volume 1 — The chain rule

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