# 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.

Source: https://practicecalculus.com/derivatives/sin-x
Updated: 2026-11-06

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

## 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:** $\int \sin x\,dx = -\cos x + C$

## Key takeaways

- The derivative of $\sin x$ is $\cos x$.
- With the chain rule, $\frac{d}{dx}\sin u = \cos u \cdot u'$, so $\frac{d}{dx}\sin(3x) = 3\cos(3x)$.
- The proof rests on two limits: $\frac{\sin h}{h} \to 1$ and $\frac{\cos h - 1}{h} \to 0$.
- The formula only holds when $x$ is measured in radians.

The **derivative of $\sin x$ is $\cos x$**. Wherever the sine wave is rising, cosine is positive. Wherever it is falling, cosine is negative.

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

## Graph of sin x and its derivative

```graph
title: 'Graph of y = sin x and its derivative y = cos x'
caption: 'At the peaks and valleys of $\sin x$ the slope is $0$, and $\cos x$ crosses the axis. At $x = 0$ the slope of $\sin x$ is $1 = \cos 0$.'
x: [-6.5, 6.5]
y: [-1.8, 1.8]
functions:
  - expr: sin(x)
    label: y = sin x
  - expr: cos(x)
    label: y = cos x (derivative)
    dashed: true
points:
  - [0, 0, 'slope 1']
  - [1.5708, 1, 'peak, slope 0']
```

## Proof from the limit definition

Use the angle sum formula $\sin(x + h) = \sin x\cos h + \cos x\sin h$:

$$\frac{\sin(x + h) - \sin x}{h} = \sin x\cdot\frac{\cos h - 1}{h} + \cos x\cdot\frac{\sin h}{h}$$

Two standard limits finish it. As $h \to 0$, $\frac{\sin h}{h} \to 1$ (proved with the [squeeze theorem](/theorems/squeeze-theorem)) and $\frac{\cos h - 1}{h} \to 0$. So

$$\frac{d}{dx}\sin x = \sin x \cdot 0 + \cos x \cdot 1 = \cos x$$

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)

$$\frac{d}{dx}\,\sin u = \cos u \cdot u'$$

Keep the inside the same, change sine to cosine, then multiply by the inside's derivative. See the [chain rule](/rules/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](/derivatives/sin-2x) and [derivative of sin(x²)](/derivatives/sin-x-squared).

## Worked examples

**Example 1: a product.** Differentiate $f(x) = x\sin x$ with the [product rule](/rules/product-rule):

$$f'(x) = \sin x + x\cos x$$

**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 $\cos\frac{\pi}{3} = \frac{1}{2}$:

$$y = \frac{\sqrt{3}}{2} + \frac{1}{2}\left(x - \frac{\pi}{3}\right)$$

**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

1. **Adding a minus sign.** $\frac{d}{dx}\sin x = \cos x$, positive. The minus belongs to the [derivative of cos x](/derivatives/cos-x).
2. **Changing the inside.** $\frac{d}{dx}\sin(3x)$ is $3\cos(3x)$, not $3\cos x$ or $\cos 3$.
3. **Confusing $\sin^2 x$ and $\sin(x^2)$.** Their derivatives are $2\sin x\cos x$ and $2x\cos(x^2)$.
4. **Working in degrees.** In degrees an extra factor $\frac{\pi}{180}$ appears.

The reverse is on the [integral of sin x](/integrals/sin-x) page. For a full overview, read [derivatives of trig functions](/blog/derivatives-of-trig-functions) or try the [trig derivative practice problems](/practice-problems/derivatives-of-trig-functions).

## Frequently asked questions

### What is the derivative of sin x?

The derivative of $\sin x$ is $\cos x$.

### What is the derivative of sin(3x)?

By the chain rule, $\cos(3x) \cdot 3 = 3\cos(3x)$.

### What is the derivative of sin x in degrees?

If $x$ is in degrees, $\frac{d}{dx}\sin x^\circ = \frac{\pi}{180}\cos x^\circ$. The clean answer $\cos x$ only works in radians, which is why calculus uses radians.

### Is the derivative of sin x equal to -cos x?

No. $\frac{d}{dx}\sin x = \cos x$. The minus sign belongs to the integral, $\int \sin x\,dx = -\cos x + C$, and to the derivative of cosine, $\frac{d}{dx}\cos x = -\sin x$.