Given the function \( g(x) = \sqrt{f(x)} \), we want to find the derivative of \( g(x) \) at \( x = 19 \), denoted as \( g'(19) \).
1. Recall the general form of the derivative of \( g(x) \):
If \( g(x) = \sqrt{f(x)} \), then using the chain rule:
[tex]\[
g'(x) = \frac{d}{dx} \left( f(x)^{1/2} \right)
\][/tex]
Applying the chain rule:
[tex]\[
g'(x) = \frac{1}{2} f(x)^{-1/2} \cdot f'(x)
\][/tex]
2. Evaluate \( f(x) \) and \( f'(x) \) at \( x = 19 \):
From the table,
[tex]\[
f(19) = 13 \quad \text{and} \quad f'(19) = -5
\][/tex]
3. Substitute these values into the derivative formula:
[tex]\[
g'(19) = \frac{1}{2} \left( f(19) \right)^{-1/2} \cdot f'(19)
\][/tex]
Substitute \( f(19) = 13 \) and \( f'(19) = -5 \):
[tex]\[
g'(19) = \frac{1}{2} \left( 13 \right)^{-1/2} \cdot (-5)
\][/tex]
4. Simplify the expression:
[tex]\[
g'(19) = \frac{1}{2} \frac{-5}{\sqrt{13}}
\][/tex]
[tex]\[
g'(19) = -\frac{5}{2 \sqrt{13}}
\][/tex]
5. Finally, compute the numerical value:
[tex]\[
g'(19) \approx -0.6933752452815365
\][/tex]
Therefore, the derivative [tex]\( g'(19) \approx -0.6933752452815365 \)[/tex].
