How to Find the Derivative of cot(x) with Respect to x | Step-by-Step Guide

(d/dx) cotx

To find the derivative of the function cot(x) with respect to x, we can use the quotient rule

To find the derivative of the function cot(x) with respect to x, we can use the quotient rule. The derivative of cot(x) can be calculated as follows:

Let y = cot(x).

1. Express cot(x) in terms of sine and cosine:
cot(x) = cos(x) / sin(x)

2. Differentiate both the numerator and denominator with respect to x:

d/dx (cos(x)) = -sin(x) (since the derivative of cos(x) is -sin(x))

d/dx (sin(x)) = cos(x) (since the derivative of sin(x) is cos(x))

3. Apply the quotient rule:

d/dx (cot(x)) = (sin(x) * d/dx(cos(x)) – cos(x) * d/dx(sin(x))) / (sin(x))^2

4. Substitute the previous derivatives into the quotient rule:

d/dx (cot(x)) = (sin(x) * (-sin(x)) – cos(x) * cos(x)) / (sin(x))^2

Simplifying further:

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

Using the Pythagorean identity sin^2(x) + cos^2(x) = 1, we can rewrite this as:

d/dx (cot(x)) = (-1) / (sin^2(x))

Therefore, the derivative of cot(x) with respect to x is:

d/dx (cot(x)) = -1 / sin^2(x)

In summary, the derivative of cot(x) with respect to x is -1/sin^2(x).

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