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Suppose \( f(x) \) and \( f^{\prime}(x) \) have the values shown.

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

Let \( g(x)=\sqrt{f(x)} \). Find \( g^{\prime}(19) \).

[tex]\[
g^{\prime}(19)= \frac{1}{2\sqrt{f(19)}} \cdot f^{\prime}(19)
\][/tex]

The value cannot be determined from the given information.

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Answer :

GinnyAnswer
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].

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