Question
Given \( f(x)=\frac{1}{\left(4-x^{2}\right)^{7}} \), find \( f^{\prime}(x) \) \( f^{\prime}(x)= \)
Ask by Coles Conner. in the United States
Jan 25,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\( f^{\prime}(x) = \frac{14x}{(4 - x^2)^8} \)
Solution
To find the derivative \( f'(x) \) of the function \( f(x) = \frac{1}{(4 - x^2)^7} \), we can use the chain rule.
1. **Rewrite the function:**
\( f(x) = (4 - x^2)^{-7} \)
2. **Differentiate using the chain rule:**
\[
f'(x) = -7 \cdot (4 - x^2)^{-8} \cdot (-2x)
\]
Here,
- The outer function is \( u^{-7} \) with derivative \( -7u^{-8} \).
- The inner function is \( u = 4 - x^2 \) with derivative \( -2x \).
3. **Simplify the expression:**
\[
f'(x) = 14x \cdot (4 - x^2)^{-8}
\]
Or equivalently,
\[
f'(x) = \frac{14x}{(4 - x^2)^8}
\]
**Final Answer:**
\[
f^{\prime}(x) = \frac{14\,x}{\,\left(4 - x^{2}\right)^{8}}
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Extra Insights
To find the derivative \( f^{\prime}(x) \) for the function \( f(x) = \frac{1}{(4 - x^2)^7} \), we can use the chain rule and the power rule. We rewrite \( f(x) \) as \( f(x) = (4 - x^2)^{-7} \). Now apply the chain rule: 1. Differentiate the outer function: \( -7(4 - x^2)^{-8} \). 2. Multiply by the derivative of the inner function: The derivative of \( 4 - x^2 \) is \( -2x \). By applying the chain rule, we have: \[ f'(x) = -7(4 - x^2)^{-8} \cdot (-2x) = 14x(4 - x^2)^{-8}. \] Thus, the derivative is: \[ f'(x) = \frac{14x}{(4 - x^2)^8}. \]
