Question

What is the derivative of `sin(pi*x)`?

Answer 1

`picospix`

Explanation:

`d/dxsinpix=picospix`

Answer 2

`dy/dx" "=" "pixxcos(u)" "=" "picos(pix)`

These days they prefer the notation:

`f^'(x)=picos(pix)` or

Explanation:

`color(blue)("Step 1")`

Set `u=pix`

`pi` is a constant so `(du)/(dx)=pi`

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Step 2")`

Set `y=sin(pix)" "->" "y=sin(u)`

Just accept that `d/(du) (sin(u))=cos(u)`

So `dy/(du)=cos(u)`

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Step 3")`

However; `" "(du)/dx xx dy/(du) " "=" " dy/dxxx(du)/(du)" " =" " dy/dx`

So by substitution we have:

`dy/dx" "=" "pixxcos(u)" "=" "picos(pix)`