Question
If \( f(x)=\frac{x}{\sqrt{x}} \), find \( f^{\prime}(25) \)
Ask by Dawson Edwards. in Malaysia
Mar 12,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( f'(25) = \frac{1}{10} \)
Solution
1. Simplify the function:
\[
f(x) = \frac{x}{\sqrt{x}} = \sqrt{x}
\]
2. Differentiate \( f(x) = \sqrt{x} = x^{\frac{1}{2}} \):
\[
f'(x) = \frac{1}{2}x^{-\frac{1}{2}} = \frac{1}{2\sqrt{x}}
\]
3. Evaluate the derivative at \( x = 25 \):
\[
f'(25) = \frac{1}{2\sqrt{25}} = \frac{1}{2 \times 5} = \frac{1}{10}
\]
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Simplify this solution Mind Expander
To find \( f^{\prime}(25) \), we first need to simplify the function \( f(x) = \frac{x}{\sqrt{x}} \). This can be rewritten as: \[ f(x) = x^{\frac{1}{2}} = x^{0.5} \] Next, we can use the power rule of differentiation to find \( f^{\prime}(x) \): \[ f^{\prime}(x) = 0.5 \cdot x^{-0.5} = \frac{0.5}{\sqrt{x}} \] Now, we can evaluate the derivative at \( x = 25 \): \[ f^{\prime}(25) = \frac{0.5}{\sqrt{25}} = \frac{0.5}{5} = 0.1 \] Thus, \[ f^{\prime}(25) = 0.1 \]
