Answer:
To determine \( f^{\prime}(x) \) for each function, we will use the power rule, chain rule, and exponential rule as necessary.
1) \( f(x) = (1-4x^3)^5 \)
To find \( f^{\prime}(x) \), we apply the chain rule. Let \( u = 1-4x^3 \), then \( f(x) = u^5 \).
Using the chain rule, we have:
\[ f^{\prime}(x) = \frac{d}{dx}(u^5) = \frac{d}{du}(u^5) \cdot \frac{du}{dx} \]
To find \( \frac{d}{du}(u^5) \), we can use the power rule:
\[ \frac{d}{du}(u^5) = 5u^{5-1} = 5u^4 \]
Now, let's find \( \frac{du}{dx} \):
[tex]\frac{du}{dx}[/tex] = [tex]\frac{d}{dx}[/tex](1-4x^3) = -12x^2 \]
Substituting these values back into the chain rule equation, we get:
\[ f^{\prime}(x) = 5u^4 \cdot (-12x^2) = -60x^2(1-4x^3)^4 \]
Therefore, \( f^{\prime}(x) = -60x^2(1-4x^3)^4 \).
The process for finding the derivatives of the remaining functions is similar, using the appropriate rules.
Copyright © 2024 EHUB.TIPS team's - All rights reserved.
Answers & Comments
Answer:
To determine \( f^{\prime}(x) \) for each function, we will use the power rule, chain rule, and exponential rule as necessary.
1) \( f(x) = (1-4x^3)^5 \)
To find \( f^{\prime}(x) \), we apply the chain rule. Let \( u = 1-4x^3 \), then \( f(x) = u^5 \).
Using the chain rule, we have:
\[ f^{\prime}(x) = \frac{d}{dx}(u^5) = \frac{d}{du}(u^5) \cdot \frac{du}{dx} \]
To find \( \frac{d}{du}(u^5) \), we can use the power rule:
\[ \frac{d}{du}(u^5) = 5u^{5-1} = 5u^4 \]
Now, let's find \( \frac{du}{dx} \):
[tex]\frac{du}{dx}[/tex] = [tex]\frac{d}{dx}[/tex](1-4x^3) = -12x^2 \]
Substituting these values back into the chain rule equation, we get:
\[ f^{\prime}(x) = 5u^4 \cdot (-12x^2) = -60x^2(1-4x^3)^4 \]
Therefore, \( f^{\prime}(x) = -60x^2(1-4x^3)^4 \).
The process for finding the derivatives of the remaining functions is similar, using the appropriate rules.