How To Differentiate Cot X With Respect To X: A Step-By-Step Guide

d/dx [cotx]

-csc^2(x)

To differentiate cot x with respect to x, we first need to express it as a function of sine and cosine using the reciprocal identity.

Recall that cot x is defined as 1/tan x or cos x/sin x. So, we can write:

cot x = cos x / sin x

Differentiating both sides with respect to x using the quotient rule, we get

d/dx[cot x] = [ (sin x) d/dx(cos x) – (cos x) d/dx(sin x) ] / (sin x)²

Now, we need to find the derivatives of sin x and cos x. By the chain rule, we get:

d/dx(sin x) = cos x

d/dx(cos x) = -sin x

Substituting into our original equation, we get:

d/dx [cot x] = [sin x(-sin x) – cos x(cos x)] / (sin x)²

Simplifying the numerator and denominator, we have:

d/dx [cot x] = (-sin² x – cos² x) / (sin²x)

Since sin² x + cos² x = 1 (the Pythagorean identity), we can simplify further:

d/dx [cot x] = -1 / (sin² x)

Therefore, the derivative of cot x with respect to x is -cosec² x or -1 / (sin²x).

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