Let $f: R \rightarrow R, g: R \rightarrow R$ and $h: R \rightarrow R$ be differentiable functions such that $f(x)=x^3+3 x+2, g(f(x))=x$ and $h(g(x))=x$ for all $x \in R$. Then, $h'(1)$ equals ________.
Answer & explanation
We have,
$g(f(x))=x$ and $h(g(g(x)))=x$ for all $x \in R$
$\Rightarrow h(g(g(f(x))))=f(x)$ for all $x \in R$ [Replacing x by f(x)]
$\Rightarrow h(g(x))=f(x)$ for all $x \in R$ [∵ g(f(x)) = x]
$\Rightarrow h(g(f(x)))=f(f(x))$ for all $x \in R$ [Replacing x by f(x)]
$\Rightarrow h(x)=f(f(x))$ for all $x \in R$ [∵ g(f(x)) = x]
$\Rightarrow h'(x)=f'(f(x)) f'(x)$ for all $x \in R$
$\Rightarrow h'(1)=f'(f(1)) f'(1)=f'(6) f'(1)$ $\left[∵ f'(x)=3 x^2+3\right]$
$\Rightarrow h'(1)=(3 \times 36+3)(3+3)=666$