Answer
\[h'(1)=\frac{6}{5}\]
Work Step by Step
$h(x)=\sqrt{4+3f(x)}$ ____(1)
Using chain rule differentiating (1) with respect to $x$
$h'(x)=\frac{1}{2\sqrt{4+3f(x)}}\cdot(4+3f(x))'$
$h'(x)=\frac{1}{2\sqrt{4+3f(x)}}\cdot [3f'(x)]$
$h'(1)=\frac{1}{2\sqrt{4+3f(1)}}\cdot [3f'(1)]$
Using given data $f(1)=7\;,\;f'(1)=4$
$h'(1)=\frac{1}{2\sqrt{4+3( 7)}}\cdot [3\times 4]$
$h'(1)=\frac{12}{2\times 5}=\frac{6}{5}$
Hence $\; h'(1)=\frac{6}{5}$.
