Given the function f, of, x, equals, minus, start fraction, square root of, x, end square root, divided by, 5, end fraction, plus, start fraction, 5, square root of, x, cubed, end square root, divided by, 6, end fraction, commaf(x)=−
5
x
​

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5
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3

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, find f, prime, of, 4, .f
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(4). Express your answer as a single fraction in simplest form.

We are given the function:

\[
f(x) = - \frac{\sqrt{x}}{5} + \frac{5 \sqrt{x^3}}{6}
\]

We need to find the derivative \( f'(x) \) and then evaluate it at \( x = 4 \).

### Step 1: Rewrite the function in a more convenient form
First, express the square roots and powers in terms of exponents:
\[
f(x) = -\frac{x^{1/2}}{5} + \frac{5x^{3/2}}{6}
\]

### Step 2: Differentiate the function
Use the power rule for differentiation, \( \frac{d}{dx} x^n = n x^{n-1} \), to find \( f'(x) \).

For the first term:
\[
\frac{d}{dx} \left( -\frac{x^{1/2}}{5} \right) = -\frac{1}{5} \cdot \frac{1}{2} x^{-1/2} = -\frac{1}{10} x^{-1/2}
\]

For the second term:
\[
\frac{d}{dx} \left( \frac{5x^{3/2}}{6} \right) = \frac{5}{6} \cdot \frac{3}{2} x^{1/2} = \frac{15}{12} x^{1/2} = \frac{5}{4} x^{1/2}
\]

Thus, the derivative is:
\[
f'(x) = -\frac{1}{10} x^{-1/2} + \frac{5}{4} x^{1/2}
\]

### Step 3: Evaluate \( f'(4) \)
Now, substitute \( x = 4 \) into \( f'(x) \):

For the first term:
\[
-\frac{1}{10} \cdot 4^{-1/2} = -\frac{1}{10} \cdot \frac{1}{2} = -\frac{1}{20}
\]

For the second term:
\[
\frac{5}{4} \cdot 4^{1/2} = \frac{5}{4} \cdot 2 = \frac{10}{4} = \frac{5}{2}
\]

Thus, \( f'(4) \) is:
\[
f'(4) = -\frac{1}{20} + \frac{5}{2}
\]

### Step 4: Combine the terms
To combine the terms, express both terms with a denominator of 20:
\[
f'(4) = -\frac{1}{20} + \frac{50}{20} = \frac{49}{20}
\]

### Final Answer:
\[
f'(4) = \frac{49}{20}
\]

Would you like any additional details or clarifications? 

Here are five related questions to explore further:
1. How do you differentiate powers of \(x\) using the power rule?
2. What is the significance of negative exponents in derivatives?
3. How do square roots convert to fractional exponents?
4. What happens if the power of \(x\) is negative when finding the derivative?
5. How would you integrate the given function instead of differentiating it?

**Tip**: When differentiating functions involving roots and exponents, always convert roots to fractional exponents for easier application of differentiation rules.

Given the function f(x) = −(√x)/5 + (5√x^3)/6, find f'(4). Express your answer as a single fraction in simplest form.

Learn how to find the derivative of the function f(x) = −(√x)/5 + (5√x^3)/6 and evaluate it at x = 4. This calculus problem involves applying the power rule and working with fractional exponents. Suitable for students in Grades 11-12.

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