To find the derivative [tex]\( f'(x) \)[/tex] of the function [tex]\( f(x) = 9 \cdot 5^{g(x)} \)[/tex] at [tex]\( x = 3 \)[/tex], we will follow these steps:
1. Express [tex]\( f(x) \)[/tex] and the given information:
[tex]\[
f(x) = 9 \cdot 5^{g(x)}
\][/tex]
Given data:
[tex]\[
g(3) = -2, \quad g'(3) = -1, \quad g''(3) = -9
\][/tex]
2. Differentiate [tex]\( f(x) \)[/tex] using the chain rule:
To find [tex]\( f'(x) \)[/tex], we apply the chain rule. First, express [tex]\( f(x) \)[/tex] as:
[tex]\[
f(x) = 9 \cdot 5^{g(x)}
\][/tex]
Taking the derivative with respect to [tex]\( x \)[/tex], we get:
[tex]\[
f'(x) = 9 \cdot \frac{d}{dx} \left( 5^{g(x)} \right)
\][/tex]
3. Differentiate [tex]\( 5^{g(x)} \)[/tex] using the chain rule and the exponential function derivative:
Recall that the derivative of [tex]\( a^{h(x)} \)[/tex] with respect to [tex]\( x \)[/tex] is [tex]\( a^{h(x)} \ln(a) h'(x) \)[/tex]. Applying this, we find:
[tex]\[
\frac{d}{dx} \left( 5^{g(x)} \right) = 5^{g(x)} \cdot \ln(5) \cdot g'(x)
\][/tex]
Hence:
[tex]\[
f'(x) = 9 \cdot 5^{g(x)} \cdot \ln(5) \cdot g'(x)
\][/tex]
4. Evaluate [tex]\( f'(x) \)[/tex] at [tex]\( x = 3 \)[/tex]:
Plug in the given values [tex]\( g(3) = -2 \)[/tex] and [tex]\( g'(3) = -1 \)[/tex]:
[tex]\[
f'(3) = 9 \cdot 5^{g(3)} \cdot \ln(5) \cdot g'(3)
\][/tex]
Substitute the known values:
[tex]\[
f'(3) = 9 \cdot 5^{-2} \cdot \ln(5) \cdot (-1)
\][/tex]
5. Simplify the expression:
[tex]\[
5^{-2} = \frac{1}{5^2} = \frac{1}{25}
\][/tex]
Therefore:
[tex]\[
f'(3) = 9 \cdot \frac{1}{25} \cdot \ln(5) \cdot (-1)
\][/tex]
[tex]\[
f'(3) = -\frac{9}{25} \cdot \ln(5)
\][/tex]
6. Approximate the numerical value:
Given the options, we see that the closest numerical value to our simplified expression of [tex]\( f'(3) \)[/tex] is approximately:
[tex]\[
f'(3) \approx -0.5794
\][/tex]
So, the value of [tex]\( f'(3) \)[/tex] is closest to:
[tex]\[
\boxed{-0.5794}
\][/tex]
