---
tag: 00R4MWW532
kind: statement
role: theorem
name: Derivative of sine
msc: [33B10, 26A24]
uses: [0WB57D9488, 002X7EK91H]
sources:
  - url: https://en.wikipedia.org/wiki/Differentiation_of_trigonometric_functions
    locator: Derivative of the sine function
---

## Statement

The sine function is differentiable on $\mathbb{R}$, and $\sin'(x) = \cos x$ for all real $x$.

## Remarks

[[Differentiate the power series|002X7EK91H]] term by term. The derivative of $\frac{(-1)^n x^{2n+1}}{(2n+1)!}$ is $\frac{(-1)^n x^{2n}}{(2n)!}$, the term of the cosine series.
With the addition formula, the claim reduces to the limits $\frac{\sin h}{h} \to 1$ and $\frac{\cos h - 1}{h} \to 0$.
Mathlib states it as `Real.hasDerivAt_sin`.

## Explanations

### L1 human

The slope of the sine curve at $x$ is $\cos x$.
At $x = 0$ the sine curve rises with slope $1$, and at $x = \pi/2$ it is flat.