Question
Find the indicated derivative. \[ f^{\prime}(x) \text { if } f(x)=3 x+\frac{5}{x} \] The derivative of the function \( f(x)=3 x+\frac{5}{x} \) is
Ask by Burgess Peterson. in the United States
Mar 09,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The derivative of \( f(x) = 3x + \frac{5}{x} \) is \( f^{\prime}(x) = 3 - \frac{5}{x^2} \).
Solution
**Step 1. Rewrite the function**
We have
\[
f(x)=3x+\frac{5}{x}.
\]
The term \(\frac{5}{x}\) can be rewritten using exponent notation as
\[
\frac{5}{x} = 5x^{-1}.
\]
So, the function becomes
\[
f(x)=3x+5x^{-1}.
\]
**Step 2. Differentiate term by term**
Differentiate \(3x\):
\[
\frac{d}{dx}(3x)=3.
\]
Differentiate \(5x^{-1}\) using the power rule \(\frac{d}{dx}(x^n)=nx^{n-1}\):
\[
\frac{d}{dx}(5x^{-1})=5\cdot(-1)x^{-1-1}=-5x^{-2}.
\]
**Step 3. Combine the results**
Thus, the derivative is
\[
f^{\prime}(x)=3-5x^{-2}.
\]
We can also express \(5x^{-2}\) as \(\frac{5}{x^2}\). Therefore, the final answer is
\[
f^{\prime}(x)=3-\frac{5}{x^2}.
\]
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Simplify this solution Extra Insights
To find the derivative \( f^{\prime}(x) \) of the function \( f(x)=3x+\frac{5}{x} \), we apply the basic differentiation rules. The derivative of \( 3x \) is \( 3 \), and to differentiate \( \frac{5}{x} \), we rewrite it as \( 5x^{-1} \), where the derivative becomes \( -5x^{-2} \). Thus, combining these results, we have: \[ f^{\prime}(x) = 3 - \frac{5}{x^2} \] So, the derivative of the function is \( f^{\prime}(x) = 3 - \frac{5}{x^2} \).
