Suppose [tex]$f(x)[tex]$[/tex] and [tex]$[/tex]f^{\prime}(x)$[/tex] have the values shown.

\begin{tabular}{|c|c|c|c|c|c|}
\hline
[tex]$x$[/tex] & 16 & 17 & 18 & 19 & 20 \\
\hline
[tex]$f(x)$[/tex] & 5 & 10 & 4 & 13 & 17 \\
\hline
[tex]$f^{\prime}(x)$[/tex] & -8 & 2 & -10 & 5 & 6 \\
\hline
\end{tabular}

Let [tex]$g(x)=\sqrt{f(x)}$[/tex].

Find [tex]$g^{\prime}(19)$[/tex].



Answer :

To find \(g'(19)\) for \(g(x) = \sqrt{f(x)}\), we need to utilize the chain rule in calculus.

Given:
[tex]\[ g(x) = \sqrt{f(x)} \][/tex]

Firstly, we need the derivative of \(g(x)\) with respect to \(x\). Using the chain rule, we obtain:
[tex]\[ g'(x) = \frac{d}{dx} \left( \sqrt{f(x)} \right) \][/tex]
[tex]\[ g'(x) = \frac{d}{dx} \left( f(x)^{1/2} \right) \][/tex]

Now, applying the chain rule:
[tex]\[ g'(x) = \frac{1}{2} f(x)^{-1/2} \cdot f'(x) \][/tex]
[tex]\[ g'(x) = \frac{1}{2 \sqrt{f(x)}} \cdot f'(x) \][/tex]

Now, we need to evaluate \(g'(x)\) at \(x=19\):
[tex]\[ g'(19) = \frac{1}{2 \sqrt{f(19)}} \cdot f'(19) \][/tex]

From the table:
[tex]\[ f(19) = 13 \][/tex]
[tex]\[ f'(19) = 5 \][/tex]

Substitute these values into the derivative formula:
[tex]\[ g'(19) = \frac{1}{2 \sqrt{13}} \cdot 5 \][/tex]

Thus:
[tex]\[ g'(19) = \frac{5}{2 \sqrt{13}} \][/tex]

To simplify:
[tex]\[ g'(19) = \frac{5}{2 \cdot 3.605551275463989} \][/tex]

Finally, we find the numerical value:
[tex]\[ g'(19) = 0.6933752452815365 \][/tex]

So, [tex]\( g'(19) \)[/tex] is approximately [tex]\( 0.6933752452815365 \)[/tex].