Problem 89E

89. If $y=f(u)$ and $u=g(x)$, where $f$ and $g$ are twice differentiable functions, show that

$$ \frac{d^{2} y}{d x^{2}}=\frac{d^{2} y}{d u^{2}}\left(\frac{d u}{d x}\right)^{2}+\frac{d y}{d u} \frac{d^{2} u}{d x^{2}} $$

Step-by-Step Solution