Question

What is the derivative of `cos(pi/x)/x^2`?

Answer 1

Quotient Rule
`d/(dx)[f(x)/(g(x))] = (g(x)f'(x) - f(x)g'(x))/(g(x))^2`

`f(x) = cos(pi/x)`
`g(x) = x^2`

The derivatives in general are:
`d/(dx)[cosu] = -sinu((du)/(dx))`
`d/(dx)[u^2] = 2u((du)/(dx))`

`=> [cancel(x^2)[-sin(pi/x)overbrace((pi)(-1/cancel(x^2)))^("Chain Rule")] - cos(pi/x)*2x]/(x^2)^2`

Simplify:
`= [pisin(pi/x) - 2xcos(pi/x)]/x^4`