Question

How do you differentiate `y=xsin(1/x)`?

Answer 1

`y'=sin(1/x)-1/xcos(1/x)`
This is a product rule + chain rule.

Explanation:

Because the inside of the sine function is something other than x, we have to do a chain rule.
So that I know what I'm doing and why, I'm going to do the chain rule first and then show how it fits into the product rule.
The derivative of `sin(1/x)` is `sin(1/x)=sin(x^-1)=cos(x^-1)(-x^-2)=cos(1/x)(-1/x^2)`

Next we'll do the product rule and put the derivative of `sin(1/x)` in there.
`y'=(1)sin(1/x)+(x)cos(1/x)(-1/x^2)`.

Now all that's left is to simplify to the final answer:
`y'=sin(1/x)-1/xcos(1/x)`.