Derivative of sin x

Q: What is the second derivative of sin x?

Differentiate again: \frac{d^2}{dx^2}\left[\sin\left(x\right)\right] = -\sin\left(x\right).

Q: What is the slope of sin x at x = 1?

Plug into the derivative: f'(1) = 0.540302. That is the slope of the tangent line there; drag the graph to see it.

\frac{d}{dx}\left[\sin\left(x\right)\right] = \cos\left(x\right)

The slope of sine traces out cosine: steepest (slope 1) where sin x crosses zero going up, flat where sine peaks. This only works in radians; in degrees an extra factor of π/180 appears.

Derivation

Trig rule
$\frac{d}{dx}\left[\sin\left(x\right)\right] = \cos\left(x\right)$

f(x) = sin x
f′(x)

Drag the graph · at x = 1.20 the slope is 0.36236, the height of the dashed curve.

Open in the derivative calculator

Questions students ask

What is the derivative of sin x?

$\frac{d}{dx}\left[\sin\left(x\right)\right] = \cos\left(x\right)$ .

What is the second derivative of sin x?

Differentiate again: $\frac{d^2}{dx^2}\left[\sin\left(x\right)\right] = -\sin\left(x\right)$ .

What is the slope of sin x at x = 1?

Plug into the derivative: $f'(1) = 0.540302$ . That is the slope of the tangent line there; drag the graph to see it.

How do you find the derivative of sin x?

The slope of sine traces out cosine: steepest (slope 1) where sin x crosses zero going up, flat where sine peaks. This only works in radians; in degrees an extra factor of π/180 appears.

What is the integral of sin x?

See the integral of sin x page for the antiderivative with steps.