Explore Westonci.ca, the leading Q&A site where experts provide accurate and helpful answers to all your questions. Join our platform to get reliable answers to your questions from a knowledgeable community of experts. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform.
Sagot :
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]
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]
We hope our answers were helpful. Return anytime for more information and answers to any other questions you may have. We hope this was helpful. Please come back whenever you need more information or answers to your queries. Westonci.ca is committed to providing accurate answers. Come back soon for more trustworthy information.
