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. Weβre, 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₯. Weβll 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. Itβs 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.
