Question

If $f\left(x\right)=\sqrt{x^2+1},g\left(x\right)=\frac{x+1}{x^2+1}$ and $h\left(x\right)=2x-3$, then find $f^{\prime }\left[h^{\prime }\left\{g^{\prime }\right(x\left)\right\}\right]$,

Answer 1

$f^{\prime }\left(x\right)=\frac{2x}{2\sqrt{x^2+1}}=\frac{x}{\sqrt{x^2+1}}$

$h^{\prime }\left(x\right)=2$

So, $h^{\prime }\left(g^{\prime }\left(x\right)\right)=2$

$f^{\prime }\left(h^{\prime }\left(g^{\prime }\left(x\right)\right)\right)=\frac{2}{\sqrt{2^2+1}}=\frac{2}{\sqrt{5}}$