Derivative of sin(2x) is 2cos(2x)

Answer

2 \cdot \mathrm{cos}\left(2 \cdot x\right)

Derivative of sin(2x)

Rule Used: Chain Rule
Steps: Identify a composite function in $\sin(2 \cdot x)$ : use the chain rule $\frac{d}{dx} f(g(x))$ = f'(g(x))·g'(x). The inner derivative is required because a stretch or shift of the input changes how fast the output moves.
Differentiate the outer function, then multiply by the inner derivative.
Final answer: $2 \cdot c$ os( $2 \cdot x$ )

The derivative of sin(2x) is 2cos(2x), with x in radians.

Instant step-by-step solution for sin(2x)

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Derivative of sin(2x) = 2*cos(2*x) Solve step-by-step for free: https://derivativecalculatorai.com/derivative-of-sin-2x

Derivative

sin(2x) → 2*cos(2*x)engine: 2*cos(2*x)

Graph of the function and its derivative

The solid curve is f. The dashed curve is f′. Vertical gaps are asymptotes or jumps, not extra zeros.

Why the derivative has this form

This is a composition, not a product: the outer function is sin(u) and the inner function is u = 2x. The chain rule says to differentiate the outer function with respect to u, keep u = 2x in the result, and multiply by du/dx. Differentiating sin(u) gives cos(u), and the derivative of 2x is 2, so the answer is 2cos(2x).

The factor 2 is easy to lose because it looks like the coefficient of a constant multiple. It is not: it comes from the rate at which the inner angle 2x changes. Doubling the angle doubles the frequency of the wave, so its peaks arrive twice as often and every slope along it is twice as steep as the corresponding slope of sin(x).

An independent check comes from the double-angle identity sin(2x) = 2 sin(x)cos(x). The product rule gives 2[cos(x)cos(x) + sin(x)(-sin(x))] = 2[cos(x)^2 - sin(x)^2] = 2cos(2x). Both routes agree, which is a useful way to catch a missing inner derivative.

Chain rule, then a product-rule cross-check

Name the layers: outer f(u) = sin(u), inner u = 2x.
Differentiate the outer function: f'(u) = cos(u).
Differentiate the inner function: du/dx = 2.
Apply the chain rule: d/dx sin(2x) = cos(u)·2 = 2cos(2x).
Check with sin(2x) = 2 sin(x)cos(x) and the product rule.
Product rule gives 2[cos(x)^2 - sin(x)^2], which is 2cos(2x) by the double-angle identity.
Sanity check at x = 0: the slope is 2cos(0) = 2, twice the slope of sin(x) at the origin.

Worked example

Slope of sin(2x) at x = pi/6

Evaluate the inner angle: 2·(pi/6) = pi/3.
The function value is sin(pi/3) = √3/2, about 0.866.
The derivative is 2cos(pi/3) = 2·(1/2) = 1.
The tangent line is y - √3/2 = 1·(x - pi/6). A secant to pi/6 + 0.001 has a slope near 1, not 0.5 or 2.

Answer: 1

Common mistakes

Answering cos(2x) and losing the 2

Differentiating only the outer sine gives cos(2x). The inner derivative du/dx = 2 must be multiplied in. At x = 0 the true slope is 2, so omitting the factor halves every slope on the curve.

Answering 2cos(x)

The angle inside the cosine stays 2x. Replacing cos(2x) with cos(x) changes the frequency of the derivative and gives a slope of 2 at x = pi/2, where sin(2x) actually has a slope of -2.

Treating sin(2x) as 2sin(x)

sin(2x) is not 2 sin(x); the correct identity is 2 sin(x)cos(x). Using the wrong identity produces 2cos(x) and drops the double-angle behaviour entirely.

Derivative of sin(x) — cos(x). This page is that result with the angle doubled.
Chain rule — The rule page lists this example and the others in the same family.
Trigonometric derivatives — The rule page lists this example and the others in the same family.
Derivatives of trigonometric functions — Same family of differentiation, different function.
Derivative of cos(2x) — Same family of differentiation, different function.
Derivative of sin^2(x) — Same family of differentiation, different function.
Derivative of cos^2(x) — Same family of differentiation, different function.
Derivative of tan^2(x) — Same family of differentiation, different function.