Derivative with arctan

Question

I need help finding the derivative of (x arctan(x^2))^-2/3

Answer 1

For this you will apply both the chain rule and the product rule..

write this as f(x)^-2/3

Use the chain rule to take the derivative of this: -2/3*f(x)^-5/3f'(x)

So the next step is to take the derivative of xarctan(x^2)

Use the product rule: u=x, v=arctan(x^2)

The derivative is u'v+uv'

u'=1

v'=1/(1+(x²)²) * 2x = 2x/(1+x⁴)

Combining everything we have that the derivative is:

-2/3*(xarctan(x²))^-5/3*(arctan(x²)+2x²/((1+x⁴))