Questions: If f(x) = (sqrt(5 * x) - 3) / (sqrt(5 * x) + 3), find f'(x). Then find f'(6).

If f(x) = (sqrt(5 * x) - 3) / (sqrt(5 * x) + 3), find f'(x). Then find f'(6).
Transcript text: If $f(x)=\frac{\sqrt{5 \cdot x}-3}{\sqrt{5 \cdot x}+3}$, find $f^{\prime}(x)$. Then find $f^{\prime}(6)$.
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Solution

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Solution Steps

Step 1: Define the Function

We start with the function defined as: \[ f(x) = \frac{\sqrt{5 \cdot x} - 3}{\sqrt{5 \cdot x} + 3} \]

Step 2: Apply the Quotient Rule

To find the derivative \( f^{\prime}(x) \), we apply the quotient rule: \[ f^{\prime}(x) = \frac{u^{\prime}(x)v(x) - u(x)v^{\prime}(x)}{(v(x))^2} \] where \( u(x) = \sqrt{5 \cdot x} - 3 \) and \( v(x) = \sqrt{5 \cdot x} + 3 \).

Step 3: Calculate the Derivative

After applying the quotient rule, we find: \[ f^{\prime}(x) = -\frac{\sqrt{5}(\sqrt{5}\sqrt{x} - 3)}{2\sqrt{x}(\sqrt{5}\sqrt{x} + 3)^2} + \frac{\sqrt{5}}{2\sqrt{x}(\sqrt{5}\sqrt{x} + 3)} \]

Step 4: Evaluate the Derivative at \( x = 6 \)

Next, we substitute \( x = 6 \) into the derivative: \[ f^{\prime}(6) = -\frac{\sqrt{30}(-3 + \sqrt{30})}{12(3 + \sqrt{30})^2} + \frac{\sqrt{30}}{12(3 + \sqrt{30})} \]

Final Answer

\(\boxed{f^{\prime}(6) = \frac{\sqrt{30}}{12(3 + \sqrt{30})}} \)

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