Video: Differentiating Inverse Trigonometric Functions

Evaluate d/dπ‘₯ cot^{βˆ’1} (1/π‘₯).

01:12

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. 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 𝑒 and 𝑒 is 𝑔 of π‘₯, then d𝑦 by dπ‘₯ is equal to d𝑦 by d𝑒 times d𝑒 by dπ‘₯. We’ll let 𝑒 be equal to one over π‘₯. Then 𝑦 is equal to the inverse cot of 𝑒. To apply the chain rule, we need to find the derivative of both of these functions. And with 𝑒 it can be useful to write it as π‘₯ to the negative one.

Then d𝑒 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 d𝑦 by d𝑒 is equal to negative one over one plus 𝑒 squared. d𝑦 by dπ‘₯ is the product of these. It’s negative one over π‘₯ squared times negative one over one plus 𝑒 squared.

We can replace 𝑒 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.

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