Derivative of sin x: Formula, Proof and Examples
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.
Trigonometric derivatives Updated 2 min read
At a glance
- Derivative
- \cos x for every real x
- Chain rule form
- \dfrac{d}{dx}\sin u = \cos u \cdot u'
- Second derivative
- -\sin x
- Units
- x must be in radians
- Integral version
- \int \sin x\,dx = -\cos x + C
On this page
The derivative of \sin x is \cos x. Wherever the sine wave is rising, cosine is positive. Wherever it is falling, cosine is negative.
Graph of sin x and its derivative
Proof from the limit definition
Use the angle sum formula \sin(x + h) = \sin x\cos h + \cos x\sin h:
Two standard limits finish it. As h \to 0, \frac{\sin h}{h} \to 1 (proved with the squeeze theorem) and \frac{\cos h - 1}{h} \to 0. So
The limit \frac{\sin h}{h} \to 1 only holds with h in radians. That is the real reason calculus uses radians.
The chain rule version: derivative of sin(u)
Keep the inside the same, change sine to cosine, then multiply by the inside's derivative. See the chain rule.
| Function | Inside u | Derivative |
|---|---|---|
| \sin(3x) | 3x | 3\cos(3x) |
| \sin(2x) | 2x | 2\cos(2x) |
| \sin(x^2) | x^2 | 2x\cos(x^2) |
| \sin(e^x) | e^x | e^x\cos(e^x) |
| \sin\left(\frac{x}{2}\right) | \frac{x}{2} | \frac{1}{2}\cos\left(\frac{x}{2}\right) |
Full pages: derivative of sin 2x and derivative of sin(x²).
Worked examples
Example 1: a product. Differentiate f(x) = x\sin x with the product rule:
Example 2: a power of sine. \frac{d}{dx}\sin^3 x = 3\sin^2 x\cos x. The outer function is the cube, the inner is \sin x.
Example 3: a tangent line. At x = \frac{\pi}{3}, \sin\frac{\pi}{3} = \frac{\sqrt{3}}{2} and the slope is \cos\frac{\pi}{3} = \frac{1}{2}:
Example 4: higher derivatives. The derivatives of \sin x cycle every four steps: \cos x, -\sin x, -\cos x, then \sin x again. So the 10th derivative is the same as the 2nd: -\sin x.
Common mistakes
- Adding a minus sign. \frac{d}{dx}\sin x = \cos x, positive. The minus belongs to the derivative of cos x.
- Changing the inside. \frac{d}{dx}\sin(3x) is 3\cos(3x), not 3\cos x or \cos 3.
- Confusing \sin^2 x and \sin(x^2). Their derivatives are 2\sin x\cos x and 2x\cos(x^2).
- Working in degrees. In degrees an extra factor \frac{\pi}{180} appears.
The reverse is on the integral of sin x page. For a full overview, read derivatives of trig functions or try the trig derivative practice problems.
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Sine, cosine and tangent
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