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

1 Answer
Jun 14, 2018

dy/dx=(arctanx)^-2(2xarctanx-x^2/(x^2+1))dydx=(arctanx)2(2xarctanxx2x2+1)

Explanation:

y=x^2/arctanxy=x2arctanx

y=x^2(arctanx)^-1y=x2(arctanx)1

Let's differentiate this using the product rule.

dy/dx=(d/dxx^2)(arctanx)^-1+x^2(d/dx(arctanx)^-1)dydx=(ddxx2)(arctanx)1+x2(ddx(arctanx)1)

The derivative of x^2x2 can be found with the product rule. To find the derivative of (arctanx)^-1(arctanx)1, use the chain rule.

dy/dx=2x(arctanx)^-1+x^2(-(arctanx)^-2)(d/dxarctanx)dydx=2x(arctanx)1+x2((arctanx)2)(ddxarctanx)

dy/dx=2x(arctanx)^-1-x^2(arctanx)^-2(1/(x^2+1))dydx=2x(arctanx)1x2(arctanx)2(1x2+1)

dy/dx=(arctanx)^-2(2xarctanx-x^2/(x^2+1))dydx=(arctanx)2(2xarctanxx2x2+1)