How to Find the Derivative of cot(x) Using the Quotient Rule | Step-by-Step Guide

(d/dx) cot(x)

To find the derivative of cot(x), we will use the quotient rule

To find the derivative of cot(x), we will use the quotient rule.

The cotangent function, cot(x), can be defined as the reciprocal of the tangent function, cot(x) = 1/tan(x).

Using the quotient rule, the derivative of cot(x) can be calculated as follows:

Let u = 1 and v = tan(x).

According to the quotient rule, the derivative of cot(x) is given by:

(d/dx) cot(x) = (v * du/dx – u * dv/dx) / (v^2)

Now, let’s calculate the derivatives of u and v:

du/dx = 0 (since u = 1, its derivative with respect to x is zero)

dv/dx = sec^2(x) (derivative of tan(x) is sec^2(x))

Plugging these values into our quotient rule formula, we have:

(d/dx) cot(x) = (tan(x) * 0 – 1 * sec^2(x)) / (tan^2(x))

Simplifying further, the derivative of cot(x) becomes:

(d/dx) cot(x) = -sec^2(x) / tan^2(x)

Recall that sec^2(x) is equal to 1 + tan^2(x), so we can substitute this in the numerator:

(d/dx) cot(x) = – (1 + tan^2(x)) / tan^2(x)

To simplify further, we can split the fraction:

(d/dx) cot(x) = -1/tan^2(x) – 1 = -csc^2(x) – 1

Therefore, the derivative of cot(x) is equal to -csc^2(x) – 1.

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