Derivative of csc x

Derivative of csc x

How to solve the derivative of csc x problems: formula, proof, example, and its solution.

Formula

[csc x]' = -csc x cot x

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

Proof

Derivative of csc x: Proof of the Formula

csc x = 1/(sin x)

Trigonometric ratio - cosecant

Reciprocal rule in differentiation

Derivative of sin x

1/(sin x) = csc x

Trigonometric ratio - cosecant

And (cos x)/(sin x) = cot x.

Trigonometric ratio - cotangent

So -1/(sin x) ⋅ (cos x)/(sin x)
= -csc x cot x.

So [csc x]' = -csc x cot x.

Example

Find the derivative of the given function. y = csc (1 - x^2)

y = csc (1 - x2)

So y' is equal to,
the derivative of the outer part, -csc (1 - x2) cot (1 - x2)
times, the derivative of the inner part, (-2x).

Chain rule in differentiation

Derivatives of polynomials