How to Find the Derivative of cot(x) with Step-by-Step Explanation

(d/dx) cot(x)

To find the derivative of cot(x), we can use the quotient rule for derivatives

To find the derivative of cot(x), we can use the quotient rule for derivatives. The quotient rule states that if you have a function in the form f(x) = g(x)/h(x), then its derivative is given by:

f'(x) = (g'(x) * h(x) – g(x) * h'(x)) / [h(x)]^2

In this case, g(x) is 1 and h(x) is tan(x). Therefore, we have:

f(x) = cot(x) = 1 / tan(x)

To find the derivative of cot(x), we need to find the derivatives of g(x) = 1 and h(x) = tan(x) and substitute them into the quotient rule.

g'(x) = 0 (the derivative of a constant is always zero)

h'(x) = sec^2(x) (the derivative of tan(x) is sec^2(x))

Substituting these values into the quotient rule formula, we get:

f'(x) = (0 * tan(x) – 1 * sec^2(x)) / [tan(x)]^2

Simplifying this expression, we have:

f'(x) = -sec^2(x) / [tan(x)]^2

Using the trigonometric identity that sec^2(x) = 1 + tan^2(x), we can rewrite the derivative as:

f'(x) = -(1 + tan^2(x)) / [tan(x)]^2

Now, we can simplify further by factoring out a -1 from the numerator:

f'(x) = -1 – tan^2(x) / [tan(x)]^2

Finally, using the trigonometric identity that 1 + tan^2(x) = sec^2(x), the derivative can be written as:

f'(x) = -sec^2(x) / [tan(x)]^2 = -1 / sin^2(x)

Therefore, the derivative of cot(x) is -1 / sin^2(x).

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