May Ortega
07/04/2024 · High School
Use the chain rule to find the derivative of \[ f(x)=8 \sqrt{3 x^{5}+8 x^{3}} \] Type your answer without fractional or negative exponents. \( f^{\prime}(x)=\square \)
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Step-by-step Solution
To find the derivative of the function
\[
f(x) = 8 \sqrt{3x^5 + 8x^3},
\]
we will use the chain rule. First, we can rewrite the function in a more convenient form:
\[
f(x) = 8 (3x^5 + 8x^3)^{1/2}.
\]
Now, we will apply the chain rule. The derivative of \( f(x) \) can be found as follows:
1. Differentiate the outer function \( 8u^{1/2} \) where \( u = 3x^5 + 8x^3 \):
\[
\frac{d}{dx}[8u^{1/2}] = 8 \cdot \frac{1}{2} u^{-1/2} \cdot \frac{du}{dx} = 4u^{-1/2} \cdot \frac{du}{dx}.
\]
2. Now, we need to find \( \frac{du}{dx} \):
\[
u = 3x^5 + 8x^3,
\]
so
\[
\frac{du}{dx} = 15x^4 + 24x^2.
\]
3. Substitute \( u \) and \( \frac{du}{dx} \) back into the derivative:
\[
f'(x) = 4(3x^5 + 8x^3)^{-1/2} \cdot (15x^4 + 24x^2).
\]
4. To express the derivative without fractional or negative exponents, we can rewrite it as:
\[
f'(x) = \frac{4(15x^4 + 24x^2)}{\sqrt{3x^5 + 8x^3}}.
\]
Thus, the final answer is:
\[
f'(x) = \frac{4(15x^4 + 24x^2)}{\sqrt{3x^5 + 8x^3}}.
\]
Quick Answer
\( f^{\prime}(x)=\frac{4(15x^4 + 24x^2)}{\sqrt{3x^5 + 8x^3}} \)
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