f ( x ) = x 2 + 1 ; g ( x ) = x + 1 x 2 + 1 ; h ( x ) = 2 x - 3 . Then the value of f ' h ' g '…

Given that

$f\left(x\right)=\sqrt{x^2+1};g\left(x\right)=\frac{x+1}{x^2+1};h\left(x\right)=2x-3$

By using Chain rule 

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

By using Quotient rule 

$g^{\text{'}}\left(x\right) = \frac{\left[x^2+1\right]\left(1\right)-\left[x\right]\left(2x\right)}{{\left(x^2+1\right)}^2} = \frac{x^3-2x^2+1}{{\left(x^2+1\right)}^2}$

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

$f^{\text{'}}\left[h^{\text{'}}\left(g^{\text{'}}\left(x\right)\right)\right] = f^{\text{'}}\left[h^{\text{'}}\left(\frac{x^3-2x^2+1}{{\left(x^2+1\right)}^2}\right)\right]$

$=f^{\text{'}}\left(2\right)$

$=\frac{2}{\sqrt{2^2+1}} = \frac{2}{\sqrt{5}}$