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

Evaluate d/d𝑥 cot^{−1} (1/𝑥).

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### Video Transcript

Evaluate the derivative of the inverse cotangent of one over 𝑥 with respect to 𝑥.

Here, we have a function of a function or a composite function. We’re, therefore, going to need to use the chain rule to find the derivative. This says that if 𝑓 and 𝑔 are differentiable functions such that 𝑦 is 𝑓 of 𝑢 and 𝑢 is 𝑔 of 𝑥, then d𝑦 by d𝑥 is equal to d𝑦 by d𝑢 times d𝑢 by d𝑥. We’ll let 𝑢 be equal to one over 𝑥. Then 𝑦 is equal to the inverse cot of 𝑢. To apply the chain rule, we need to find the derivative of both of these functions. And with 𝑢 it can be useful to write it as 𝑥 to the negative one.

Then d𝑢 by d𝑥 is negative 𝑥 to negative two or negative one over 𝑥 squared. We can then use the general derivative of the inverse cotangent function. And we see that d𝑦 by d𝑢 is equal to negative one over one plus 𝑢 squared. d𝑦 by d𝑥 is the product of these. It’s negative one over 𝑥 squared times negative one over one plus 𝑢 squared.

We can replace 𝑢 with one over 𝑥 and then multiply through. And we see that the derivative of the inverse cotangent of one over 𝑥 with respect to 𝑥 is one over 𝑥 squared plus one.