Question

Find the derivative of the function y=x² /arctgx ?

Answer 1

`dy/dx=(arctanx)^-2(2xarctanx-x^2/(x^2+1))`

Explanation:

`y=x^2/arctanx`

`y=x^2(arctanx)^-1`

Let's differentiate this using the product rule.

`dy/dx=(d/dxx^2)(arctanx)^-1+x^2(d/dx(arctanx)^-1)`

The derivative of `x^2` can be found with the product rule. To find the derivative of `(arctanx)^-1`, use the chain rule.

`dy/dx=2x(arctanx)^-1+x^2(-(arctanx)^-2)(d/dxarctanx)`

`dy/dx=2x(arctanx)^-1-x^2(arctanx)^-2(1/(x^2+1))`

`dy/dx=(arctanx)^-2(2xarctanx-x^2/(x^2+1))`