If f(x) + x^2[f(x)]^5 = 34 and f(1) = 2, find f '(1).

Question

If f(x) + x²[f(x)]⁵=34 and f(1) = 2,find f '(1)



f'(1)=______

Answer 1

Let y = f(x)    

The problem is then a standard implicit differentiation problem.

y + x^2 y^5 = 34

differentiate both side wrt x:

 y’ + 2x y^5 + 5 x^2 y^4 y’ = 0

Solve for y’

y’ = -2 x y^5 /(1 + 5 x^2 y^4)

Plug in y = 2 and x = 1

y’ = -64/81