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.