Math · Calculus I · Worked example

Differentiate sin(x²) and sin²x

Differentiate y = sin(x²) and y = sin²x, and explain why the two answers differ.

sin (x^{2}) and {sin}^{2} x

Find the layers of sin(x²)

Work out sin(x²) for x = 3: first square 3 to get 9, then take sin 9. The squaring happens first, so it is the inside. The sine happens last, so it is the outside.

u = x^{2}, y = sin u

Differentiate each layer

The outside, sin u, has derivative cos u with respect to u. The inside, x², has derivative 2x with respect to x.

\frac{d y}{d u} = cos u, \frac{d u}{d x} = 2 x

Multiply, then put the inside back

Multiply the two rates, then replace u with x² so the answer is in terms of x. Writing the 2x in front is the usual form.

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

Find the layers of sin²x

sin²x is shorthand for (sin x)². For x = 3, you first find sin 3 and then square the result. Now the sine is the inside and the square is the outside, the reverse of sin(x²).

u = sin x, y = u^{2}

Differentiate each layer

The outside, u², has derivative 2u. The inside, sin x, has derivative cos x.

\frac{d y}{d u} = 2 u, \frac{d u}{d x} = cos x

Multiply, then put the inside back

Multiply to get 2u·cos x, then replace u with sin x.

\frac{d y}{d x} = 2 sin x cos x

Compare the two answers

Both functions are built from a sine and a square, in opposite orders, and the chain rule follows the order. Whichever operation is done last is the outside: its derivative is the one evaluated at the inside. The other operation supplies the multiplying factor.

Result

The derivative of sin(x²) is 2x cos(x²), and the derivative of sin²x is 2 sin x cos x.

\begin{array}{l} \frac{d}{d x} sin (x^{2}) = 2 x cos (x^{2}) \\ \frac{d}{d x} {sin}^{2} x = 2 sin x cos x \end{array}

Your turn

Differentiate y = cos(x³).

Show the answer and explanation

dy/dx = −3x² sin(x³).

The inside is u = x³ and the outside is cos u. The outside gives −sin u, evaluated at the inside as −sin(x³). The inside gives 3x². Multiply: −sin(x³)·3x² = −3x² sin(x³).

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

Keep exploring

Open both derivatives in Math: the Checker marks each one ✓. Change 2x cos(x²) to cos(x²) and its mark turns to ✗, because the inside’s factor 2x is missing.

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OpenStax Calculus Volume 1 — The chain rule

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