Question

What is the derivative of this function `y = x sin (5/x)`?

Answer 1

`dy/dx=sin(5/x)-5/xcos(5/x)`

Explanation:

First we need the product rule, which states that `d/dx(uv)=((du)/dx)v+u((dv)/dx)`. Thus:

`dy/dx=(d/dxx)sin(5/x)+x(d/dxsin(5/x))`

Here, `d/dxx=1`.

To figure out `d/dxsin(5/x)`, we need the chain rule since we have a function inside another function. Knowing that `d/dxsin(x)=cos(x)`, we see that through the chain rule, `d/dxsin(u)=cos(u)((du)/dx)`.

Then:

`dy/dx=sin(5/x)+xcos(5/x)(d/dx(5/x))`

Note that `d/dx5/x=d/dx5x^-1=-5x^-2` through the power rule:

`dy/dx=sin(5/x)+xcos(5/x)*(-5/x^2)`

`dy/dx=sin(5/x)-5/xcos(5/x)`