If f is a function such that $\displaystyle 2f(x)+x(f(x))^3=x^2$, find the value of $\displaystyle f'(2)$
By the chain rule(and product rule):
$\displaystyle 2 f(x) + x(f(x))^3 = x^2 \implies 2 f'(x) + (f(x))^3 + 3x(f(x))^2 \cdot f'(x) = 2x$
now solve for $\displaystyle f'(x)$ and you can find $\displaystyle f'(2)$
note, you will need the value for $\displaystyle f(2)$, but no worries, you can find that from the original equation given.