Answer
$$\frac{-16}{13}$$
Work Step by Step
Given $$f(x) +x^2[f(x)]^3=10$$
Differentiate the both sides
\begin{align*}
f'(x)+3x^2[f(x)]^2f'(x)+2x[f(x)]^3&=0\\
f'(x) (1+3x^2[f(x)]^2 )&=-2x[f(x)]^3\\
f'(x) &= \frac{-2x[f(x)]^3}{1+3x^2[f(x)]^2}\\
\end{align*}
Since $f(1)= 2$, then
\begin{align*} f'(1) &= \frac{-2(1)[f(1)]^3}{1+3(1)^2[f(1)]^2}\\
&= \frac{-2(8)}{1+3(4)}\\
&=\frac{-16}{13}
\end{align*}
