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].
