Derivative of cot x

Derivative of cot x

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

Formula

[cot x]' = -csc^2 x

The derivative of cot x is -csc2 x.

Proof

Derivative of cot x: Proof of the Formula

cot x = 1/(tan x)

Trigonometric ratio - cotangent

Reciprocal rule in differentiation

Derivative of tan x

sec2 x = 1/(cos2 x)

Trigonometric ratio - secant

And 1/(tan2 x) = (cos2 x)/(sin2 x).

Trigonometric ratio - tangent

Cancel cos2 x.

1/(sin x) = csc x

Trigonometric ratio - cosecant

So -1/(sin2 x) = -csc2 x.

So [cot x]' = -csc2 x.

Example

Find the derivative of the given function. y = 7x cot x

y = (7x)(tan x)

So y' is equal to,
the derivative of 7x, 7
times cot x
plus 7x,
times, the derivative of cot x, -csc2 x.

Product rule in differentiation

Power rule in differentiation (Part 1)