If d/dx[f(x^3)]=-6x^4 what is f'(x)

Question

Please help, I don't know how to find f'(x) given the information.

Answer 1

Hi, Shrimp. Let's do this one!

If d/dx (f(x³)) = - 6x⁴, then f(x³) = -6x⁵ / 5 + C . . . . . taking the anti-derivative, for some constant, C

and f(x) = -6x^5/3 / 5 + C . . . . replacing x by x³ in this equation, gives the one above

∴ f '(x) = (-6/5) (5/3) x^{5/3 - 1}

= -2 x^2/3