Single Variable Calculus, Chapter 3, 3.6, Section 3.6, Problem 21

Suppose that $f(x) + x^2 [f(x)]^3 = 10$ and $f(1) = 2$, find $f'(1)$
$\displaystyle \frac{d}{dx} [ f(x) ] + \frac{d}{dx} \left( x^2 [f(x)]^3 \right) = \frac{d}{dx}(10)$

$
\begin{equation}
\begin{aligned}
\frac{d}{dx} [f(x)] + \left[ (x^2) \frac{d}{dx} [f(x)]^3 + [f(x)]^3 \frac{d}{dx} (x^2) \right] &= \frac{d}{dx} (10)\\
\\
f'(x) + (x^2) (3) [f(x)]^2 \frac{d}{dx} [f(x)] + [f(x)]^3 (2x) &= 0\\
\\
f'(x) + 3x^2 [f(x)]^2 f'(x) + 2x [f(x)]^3 &= 0\\
\\
\end{aligned}
\end{equation}
$

when $x = 1$,

$
\begin{equation}
\begin{aligned}
f'(1) + 3(1)^2 [f(1)]^2 f'(1) + 2(1) [f(1)]^3 &= 0\\
\\
f'(1) + 3(2)^2 f'(1) + 2(2)^3 &= 0\\
\\
f'(1) + 12f'(1) + 16 &= 0\\
\\
13f'(1) + 16 &= 0\\
\\
13'f(1) &= -16\\
\\
\frac{\cancel{13}f'(1)}{\cancel{13}} &= \frac{-16}{13}\\
\\
f'(1) &= \frac{-16}{13}
\end{aligned}
\end{equation}
$