Question

Math- Differentiation . Could you help me to solve this question?

Answer 1

Explanation:

Hello, first of all we can find a value for f(1)

`xf(x)+f(x^2)=2 \\\\\text{So for x = 1, it gives}\\\\f(1)+f(1^2)=f(1)+f(1)=2f(1)=2\\\\<=> f(1) =1`

And we can get the derivative of the equation so.

`(uv)'=u'v+uv' \text{ and } (df(x^2))/(dx)=2xf'(x^2) \text{ so we can write}\\\\f(x)+xf'(x)+2xf'(x^2)=0\\\\\text{And then, for x = 1}\\\\f(1)+f'(1)+2f'(1)=0\\\\<=> f(1)+3f'(1)=0\\\\<=> 3f'(1)=-f(1)=-1\\\\<=>\large \boxed{ f'(1)=-(1)/(3) }`

Thank you