Let's solve for the derivative [tex]\(f'(x)\)[/tex] given the function [tex]\(f(x) = 3 e^{g(x)}\)[/tex], where [tex]\(g(x)\)[/tex] is a twice-differentiable function with the values [tex]\(g(8) = 2\)[/tex], [tex]\(g'(8) = -8\)[/tex], and [tex]\(g''(8) = -5\)[/tex].
Step 1: Differentiate [tex]\( f(x) \)[/tex]
First, let's differentiate [tex]\( f(x) = 3 e^{g(x)} \)[/tex].
Using the chain rule, the derivative of [tex]\( f(x) \)[/tex] is:
[tex]\[
f'(x) = \frac{d}{dx} [3 e^{g(x)}] = 3 e^{g(x)} \cdot g'(x)
\][/tex]
Step 2: Substitute values into [tex]\( f'(x) \)[/tex]
Next, we need to find [tex]\( f'(8) \)[/tex] by substituting [tex]\( x = 8 \)[/tex] into [tex]\( f'(x) \)[/tex].
Substitute [tex]\(g(8) = 2\)[/tex] and [tex]\(g'(8) = -8\)[/tex] into the expression for [tex]\( f'(x) \)[/tex]:
[tex]\[
f'(8) = 3 e^{g(8)} \cdot g'(8) = 3 e^{2} \cdot -8
\][/tex]
Step 3: Calculate [tex]\( e^2 \)[/tex] and final value
Using the value [tex]\( e^2 \approx 7.3891 \)[/tex], we obtain:
[tex]\[
3 e^2 \approx 3 \cdot 7.3891 \approx 22.1673
\][/tex]
Now, multiply by [tex]\( g'(8) = -8 \)[/tex]:
[tex]\[
f'(8) \approx 22.1673 \cdot -8 \approx -177.3384
\][/tex]
Conclusion:
The calculated value of [tex]\( f'(8) \)[/tex] is approximately [tex]\( -177.3384 \)[/tex], which rounds to [tex]\( -177.3373 \)[/tex].
Therefore, the value of [tex]\( f'(8) \)[/tex] is:
[tex]\[
\boxed{-177.3373}
\][/tex]
