# Question Video: Differentiating Inverse Trigonometric Functions Mathematics • Higher Education

Evaluate d/dπ₯ cot^{β1} (1/π₯).

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### Video Transcript

Evaluate the derivative of the inverse cotangent of one over π₯ with respect to π₯.

Here, we have a function of a function or a composite function. Weβre, therefore, going to need to use the chain rule to find the derivative. This says that if π and π are differentiable functions such that π¦ is π of π’ and π’ is π of π₯, then dπ¦ by dπ₯ is equal to dπ¦ by dπ’ times dπ’ by dπ₯. Weβll let π’ be equal to one over π₯. Then π¦ is equal to the inverse cot of π’. To apply the chain rule, we need to find the derivative of both of these functions. And with π’ it can be useful to write it as π₯ to the negative one.

Then dπ’ by dπ₯ is negative π₯ to negative two or negative one over π₯ squared. We can then use the general derivative of the inverse cotangent function. And we see that dπ¦ by dπ’ is equal to negative one over one plus π’ squared. dπ¦ by dπ₯ is the product of these. Itβs negative one over π₯ squared times negative one over one plus π’ squared.

We can replace π’ with one over π₯ and then multiply through. And we see that the derivative of the inverse cotangent of one over π₯ with respect to π₯ is one over π₯ squared plus one.