From simple questions to complex issues, IDNLearn.com has the answers you need. Whether it's a simple query or a complex problem, our experts have the answers you need.
Sagot :
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]
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]
Thank you for using this platform to share and learn. Keep asking and answering. We appreciate every contribution you make. IDNLearn.com has the solutions to your questions. Thanks for stopping by, and see you next time for more reliable information.