Macdonald Bird
06/01/2024 · Middle School
If \( f(x)=(3 x+5)^{-4} \), find \( f^{\prime}(x) \) Find \( f^{\prime}(3) \) .
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Tutor-Verified Answer
Step-by-step Solution
To find the derivative \( f'(x) \) of the function \( f(x) = (3x + 5)^{-4} \), we will use the chain rule.
The chain rule states that if you have a composite function \( g(h(x)) \), then the derivative is given by:
\[
g'(h(x)) \cdot h'(x)
\]
In our case, let \( g(u) = u^{-4} \) where \( u = 3x + 5 \).
First, we find the derivative of \( g(u) \):
\[
g'(u) = -4u^{-5}
\]
Next, we find the derivative of \( h(x) = 3x + 5 \):
\[
h'(x) = 3
\]
Now, applying the chain rule:
\[
f'(x) = g'(h(x)) \cdot h'(x) = -4(3x + 5)^{-5} \cdot 3
\]
Simplifying this gives:
\[
f'(x) = -12(3x + 5)^{-5}
\]
Now, we need to find \( f'(3) \):
First, we calculate \( 3x + 5 \) at \( x = 3 \):
\[
3(3) + 5 = 9 + 5 = 14
\]
Now substituting \( x = 3 \) into \( f'(x) \):
\[
f'(3) = -12(14)^{-5}
\]
Calculating \( 14^{-5} \):
\[
14^{-5} = \frac{1}{14^5}
\]
Thus,
\[
f'(3) = -12 \cdot \frac{1}{14^5} = -\frac{12}{14^5}
\]
Finally, we can express \( 14^5 \):
\[
14^5 = 537824
\]
So,
\[
f'(3) = -\frac{12}{537824}
\]
Thus, the final answer is:
\[
f'(x) = -12(3x + 5)^{-5}, \quad f'(3) = -\frac{12}{537824}
\]
Quick Answer
\( f'(x) = -12(3x + 5)^{-5} \), \( f'(3) = -\frac{12}{537824} \)
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