Calculus • Step-by-Step

Derivative of csc(x) with Steps

Differentiate csc(x) step by step using the quotient rule or reciprocal identity 1/sin(x) to prove that d/dx[csc(x)] = -csc(x)cot(x) with full steps.

Problem Statement

\frac{d}{dx}[\csc(x)]

Final Result

f'(x) = -\frac{\mathrm{cos}\left(x\right)}{\mathrm{sin}\left(x\right)^{2}}

Formula Required: Cosecant Derivative Rule

Differentiating (sin x)^(-1) yields -(sin x)^(-2) · cos x = -cos x / sin² x = -csc x cot x.

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

2 Steps

Set up the differentiation problem with respect to variable $x$.

\frac{d}{dx} \left[ \mathrm{csc}\left(x\right) \right]

Apply the Power Rule to compute the derivative.

\frac{d}{dx} \left[ \mathrm{csc}\left(x\right) \right] = -\frac{\mathrm{cos}\left(x\right)}{\mathrm{sin}\left(x\right)^{2}}

Interactive Calculator

Enter any equation like 2x + 8 = 20 or x² - 5x + 6 = 0 to get step-by-step solutions.

The derivative of csc(x) is -csc(x)cot(x).