Derivative of csc x

Q: What is the second derivative of csc x?

Differentiate again: \frac{d^2}{dx^2}\left[\csc\left(x\right)\right] = \cot^{2}\left(x\right) \csc\left(x\right) + \csc^{3}\left(x\right).

Q: What is the slope of csc x at x = 1.8?

Plug into the derivative: f'(1.8) = 0.239569. That is the slope of the tangent line there; drag the graph to see it.

\frac{d}{dx}\left[\csc\left(x\right)\right] = -\cot\left(x\right) \csc\left(x\right)

csc x = 1/sin x, so the chain rule gives −cos x / sin²x, which is written −csc x · cot x. The derivative is negative where csc x is positive and the curve is falling toward its minimum at π/2.

Derivation

Trig rule
$\frac{d}{dx}\left[\csc\left(x\right)\right] = -\cot\left(x\right) \csc\left(x\right)$

f(x) = csc x
f′(x)

Drag the graph · at x = 1.96 the slope is 0.44328, the height of the dashed curve.

Open in the derivative calculator

Questions students ask

What is the derivative of csc x?

$\frac{d}{dx}\left[\csc\left(x\right)\right] = -\cot\left(x\right) \csc\left(x\right)$ .

What is the second derivative of csc x?

Differentiate again: $\frac{d^2}{dx^2}\left[\csc\left(x\right)\right] = \cot^{2}\left(x\right) \csc\left(x\right) + \csc^{3}\left(x\right)$ .

What is the slope of csc x at x = 1.8?

Plug into the derivative: $f'(1.8) = 0.239569$ . That is the slope of the tangent line there; drag the graph to see it.

How do you find the derivative of csc x?

csc x = 1/sin x, so the chain rule gives −cos x / sin²x, which is written −csc x · cot x. The derivative is negative where csc x is positive and the curve is falling toward its minimum at π/2.

What is the integral of csc x?

Use the integral calculator to find the antiderivative of csc x step by step.