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 (x). Express your answer in radical form without using negative exponents, simplifying all fractions.

Given the function \( f(x) = \frac{5}{6 \sqrt{x}} \), we want to find the derivative \( f'(x) \).

### Step 1: Rewrite the function in a more derivative-friendly form
The square root of \( x \) can be expressed as \( x^{1/2} \), and the denominator can be simplified using this. So:

\[
f(x) = \frac{5}{6} x^{-\frac{1}{2}}
\]

### Step 2: Apply the power rule for differentiation
The power rule states that for \( x^n \), the derivative is \( n x^{n-1} \). In this case, the function is of the form \( ax^n \), where \( a = \frac{5}{6} \) and \( n = -\frac{1}{2} \).

Now, differentiating:

\[
f'(x) = \frac{5}{6} \cdot \left(-\frac{1}{2}\right) x^{-\frac{1}{2}-1}
\]

### Step 3: Simplify the expression
\[
f'(x) = \frac{5}{6} \cdot \left(-\frac{1}{2}\right) x^{-\frac{3}{2}}
\]
\[
f'(x) = -\frac{5}{12} x^{-\frac{3}{2}}
\]

### Step 4: Convert back to radical form
Since \( x^{-\frac{3}{2}} = \frac{1}{x^{\frac{3}{2}}} = \frac{1}{x \sqrt{x}} \), we can rewrite the derivative as:

\[
f'(x) = -\frac{5}{12} \cdot \frac{1}{x \sqrt{x}}
\]

Thus, the final answer in radical form is:

\[
f'(x) = -\frac{5}{12 x \sqrt{x}}
\]

Would you like more details or have any questions?

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Here are 5 related questions to expand on this concept:

1. How do you differentiate a function like \( f(x) = \frac{7}{8 \sqrt{x}} \)?
2. What is the second derivative of \( f(x) = \frac{5}{6 \sqrt{x}} \)?
3. How do you find the derivative of \( f(x) = \frac{5}{6} x^{3/2} \)?
4. What happens if the original function is \( f(x) = \frac{5}{6 x} \) instead of involving a square root?
5. How do you apply the chain rule if the function inside the square root was more complicated?

### Tip:
Always rewrite functions involving roots and fractions in terms of exponents to simplify differentiation.

Given the function f(x) = 5 / (6√x), find f'(x). Express your answer in radical form without using negative exponents, simplifying all fractions.

This page provides a step-by-step solution to find the derivative of the function f(x) = 5 / (6√x) using the power rule. The solution simplifies the expression and converts it back into radical form. Ideal for students in Grades 11-12 learning calculus concepts such as differentiation and radical expressions.

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