How to Find the Derivative of Cot x | A Step-by-Step Guide

d/dx [cot x]

To find the derivative of cot x, we can start by expressing it in terms of sine and cosine:

cot x = cos x / sin x

Now, let’s differentiate both sides with respect to x using the quotient rule:

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

Using the quotient rule, the derivative of the numerator, cos x, is -sin x, and the derivative of the denominator, sin x, is cos x

To find the derivative of cot x, we can start by expressing it in terms of sine and cosine:

cot x = cos x / sin x

Now, let’s differentiate both sides with respect to x using the quotient rule:

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

Using the quotient rule, the derivative of the numerator, cos x, is -sin x, and the derivative of the denominator, sin x, is cos x. Applying the quotient rule:

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

Simplifying the numerator:

(d/dx) [cot x] = [-sin² x – cos² x] / (sin x)²

Recall the Pythagorean identity, which states that sin² x + cos² x = 1. Rearranging this equation:

cos² x = 1 – sin² x

Substituting this identity into the numerator:

(d/dx) [cot x] = [-(1 – sin² x) – cos² x] / (sin x)²

(d/dx) [cot x] = [-1 + sin² x – cos² x] / (sin x)²

Since sin² x + cos² x = 1, we have:

(d/dx) [cot x] = [-1 + 1] / (sin x)²

Simplifying further:

(d/dx) [cot x] = 0 / (sin x)²

The final result is:

(d/dx) [cot x] = 0

Therefore, the derivative of cot x with respect to x is 0.

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