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Trigonometric derivatives

Derivative of sin 2x: Formula, Proof and Examples

The derivative of sin 2x is 2cos 2x. See the chain rule proof, a check with 2 sin x cos x, the general sin(ax) rule and worked examples.

Trigonometric derivatives Updated 2 min read

At a glance

\frac{d}{dx}\,\sin 2x = 2\cos 2x
Derivative
2\cos 2x
General rule
\dfrac{d}{dx}\sin(ax) = a\cos(ax)
Second derivative
-4\sin 2x
Period
\pi, half the period of \sin x
Integral of sin 2x
\int\sin 2x\,dx = -\frac{1}{2}\cos 2x + C
On this page
  1. Graph of sin 2x and its derivative
  2. Proof with the chain rule
  3. A check with the double angle formula
  4. Why the 2 comes out
  5. Worked examples
  6. Where you will meet it
  7. Common mistakes

The derivative of \sin 2x is 2\cos 2x. The chain rule pulls the inner factor 2 out front, and the inside 2x stays the same.

\frac{d}{dx}\,\sin 2x = 2\cos 2x

Graph of sin 2x and its derivative

Graph of y = sin 2x and its derivative y = 2cos 2x
\sin 2x oscillates twice as fast as \sin x, so its slopes are twice as steep. The derivative reaches \pm 2.

Proof with the chain rule

Outer function \sin u, inner function u = 2x, so u' = 2. With the chain rule and the derivative of sin x:

\frac{d}{dx}\sin(2x) = \cos(2x)\cdot 2 = 2\cos 2x

A check with the double angle formula

Rewrite \sin 2x = 2\sin x\cos x and use the product rule:

\frac{d}{dx}(2\sin x\cos x) = 2\left(\cos x\cdot\cos x + \sin x\cdot(-\sin x)\right) = 2(\cos^2 x - \sin^2 x)

The double angle formula \cos^2 x - \sin^2 x = \cos 2x turns this into 2\cos 2x. Two methods, one answer.

Why the 2 comes out

Squeezing the graph horizontally by a factor of 2 makes it twice as steep everywhere. That is what the factor 2 in front of \cos 2x records. In general:

\frac{d}{dx}\sin(ax) = a\cos(ax), \qquad \frac{d}{dx}\sin(ax + b) = a\cos(ax + b)
Function Derivative
\sin 3x 3\cos 3x
\sin\frac{x}{2} \frac{1}{2}\cos\frac{x}{2}
\sin(2x + 1) 2\cos(2x + 1)
\sin(\pi x) \pi\cos(\pi x)
5\sin 2x 10\cos 2x

Worked examples

Example 1: a value. At x = \frac{\pi}{6}, the slope is 2\cos\frac{\pi}{3} = 2\cdot\frac{1}{2} = 1.

Example 2: a product. \frac{d}{dx}(x\sin 2x) = \sin 2x + 2x\cos 2x.

Example 3: higher derivatives. Each derivative brings out another 2: \frac{d^2}{dx^2}\sin 2x = -4\sin 2x, \frac{d^3}{dx^3}\sin 2x = -8\cos 2x, and \frac{d^4}{dx^4}\sin 2x = 16\sin 2x.

Example 4: a link to sin²x. The derivative of sin²x is \sin 2x. So \sin^2 x is an antiderivative of \sin 2x, and so is -\frac{1}{2}\cos 2x. They differ by a constant, since \sin^2 x = \frac{1}{2} - \frac{1}{2}\cos 2x.

Where you will meet it

The function \sin 2x appears whenever a wave repeats twice as often as \sin x. It also shows up as a simplified answer, for example as the derivative of \sin^2 x, so knowing its own derivative saves time on second derivatives. In physics, \sin 2\theta appears in the range of a projectile, and its derivative tells you how the range changes as you tilt the launch angle.

Common mistakes

  1. Forgetting the 2. \frac{d}{dx}\sin 2x is not \cos 2x.
  2. Changing the inside. The answer is 2\cos 2x, not 2\cos x.
  3. Dividing instead of multiplying. \frac{1}{2} shows up when you integrate, not when you differentiate. See the integral of sin 2x.
  4. Confusing \sin 2x with 2\sin x. \frac{d}{dx}(2\sin x) = 2\cos x. The graphs are different.

For more chain rule drills, try the chain rule practice problems.

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Chain rule: trig

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Frequently asked questions

What is the derivative of sin 2x?
It is 2\cos 2x, by the chain rule.

Why is the derivative of sin 2x not cos 2x?
Because of the chain rule. The inside 2x has derivative 2, and you must multiply by it.

What is the derivative of 2 sin x cos x?
It is 2\cos 2x, since 2\sin x\cos x = \sin 2x. The product rule gives 2(\cos^2 x - \sin^2 x), which is the same.

What is the derivative of sin(2x + 1)?
It is 2\cos(2x + 1). The constant 1 does not change the inner derivative.

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Related derivatives

Derivative of sin x \frac{d}{dx}\,\sin x = \cos x The derivative of sin x is cos x. See the limit proof, the chain rule version for sin(u), a graph and worked examples like sin(3x) and x sin x. Derivative of cos x \frac{d}{dx}\,\cos x = -\sin x The derivative of cos x is -sin x. See the limit proof, a quick proof from sin x, the chain rule for cos(u) and worked examples like cos(5x). Derivative of sin²x \frac{d}{dx}\,\sin^2 x = 2\sin x\cos x = \sin 2x The derivative of sin²x is 2 sin x cos x, which equals sin 2x. See the chain rule proof, a check with a double angle identity and examples.