Derivative/(g(f(x)))^2f(g(x))/f is x^2-1 and g is x+2/Exercise

Let

$f,g\colon \mathbb {R} \longrightarrow \mathbb {R}$

be two differentiable functions and consider

${}h(x)=(g(f(x)))^{2}f(g(x))\,.$

a) Determine the derivative ${}h'$ from the derivatives of ${}f$ and ${}g$. b) Let now

$f(x)=x^{2}-1{\text{ and }}g(x)=x+2.$

Compute ${}h'(x)$ in two ways, one directly from ${}h(x)$ and the other by the formula of part ${}a)$.

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