Question

What is the derivative of `( cos (pi*x) +1 ) / x`?

Answer 1

`d/dx (cos(pi x)+1)/x= -(pisin(pix))/x - (cos(pix))/x^2`

Explanation:

If you are studying maths, then you should learn the Quotient Rule for Differentiation, and practice how to use it:

`d/dx(u/v) = (v(du)/dx-u(dv)/dx)/v^2`, or less formally, `(u/v)' = (v(du)-u(dv))/v^2`

I was taught to remember the rule in word; " vdu minus udv all over v squared ". To help with the ordering I was taught to remember the acronym, VDU as in Visual Display Unit.

So Let `y=` we have;

`dy/dx = ( (x)(d/dxcos pi x) - (cos pi x)(d/dxx) ) / (x)^2`
`:. dy/dx = ( (x)(-pisinpix) - (cos pi x)(1) ) / x^2`
`:. dy/dx = ( -pixsinpix - cos pi x ) / x^2`
`:. dy/dx = -(pisinpix)/x - (cos pi x)/x^2`