Question

What is the derivative of `arctan(x-1)`?

Answer 1

`d/dx(arctan(x-1))=1/(x^2+2x+2)`

Explanation:

We must first know the derivative of `arctan(x)`, which is:

`d/dxarctan(x)=1/(1+x^2)`

According to the chain rule, we see that

`d/dxarctan(f(x))=1/(1+(f(x))^2)*f'(x)`

So, for `arctan(x-1)`, we see that `f(x)=x-1` and `f'(x)=1`.

`d/dxarctan(x-1)=1/(1+(x-1)^2)*1`

`=1/(1+(x^2-2x+1)`

`=1/(x^2-2x+2)`