Finding the Derivative of the Function 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 start by rewriting cot(x) as 1/tan(x)

To find the derivative of the function cot(x) with respect to x, we can start by rewriting cot(x) as 1/tan(x).

Now, let’s differentiate 1/tan(x) using the quotient rule. The quotient rule states that if we have a function 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) = 1 and h(x) = tan(x).

Differentiating g(x) = 1 with respect to x gives us g'(x) = 0, as any constant function has a derivative of zero.

Differentiating h(x) = tan(x) with respect to x can be done using the chain rule. The derivative of tan(x) is sec^2(x), multiplied by the derivative of the inner function x.

So, h'(x) = sec^2(x) * 1 = sec^2(x).

Now, let’s substitute these values into the quotient rule formula:

f'(x) = (0 * tan(x) – 1 * sec^2(x)) / (tan(x))^2
= -sec^2(x) / tan^2(x).

However, we can simplify this expression further using trigonometric identities. Recall that sec^2(x) = 1/cos^2(x) and tan^2(x) = sin^2(x) / cos^2(x).

So, f'(x) = -1 / (cos^2(x) * (sin^2(x) / cos^2(x)))
= -1 / sin^2(x)
= -csc^2(x).

Therefore, the derivative of cot(x) with respect to x is -csc^2(x).

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