Question

What is the derivative of `tan(1/x)`?

Answer 1

`= - sec^2 (1/x) * (1/x^2)`

Explanation:

we use the chain rule

to make it clear we can say let `w = 1/x`

so `y = tan w` means that `dy/(dw) = sec^2 w`

and `(dw)/dx = d/dx (1/x) = - 1/x^2`

the chain rule tells us that

`dy/dx = dy/(dw) * (dw)/dx`

`= sec^2 w * (- 1/x^2)`

`= sec^2 (1/x) * (- 1/x^2)`

`= - sec^2 (1/x) * (1/x^2)`