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We are tasked with finding the derivative of the function:

\[
f(x) = \frac{-20}{5\sqrt{8x+5}}
\]

### Step 1: Simplify the function

We can rewrite the given function as:

\[
f(x) = \frac{-4}{\sqrt{8x + 5}}
\]

This is because \(\frac{-20}{5} = -4\).

### Step 2: Apply the chain rule

The function is of the form:

\[
f(x) = -4 \cdot (8x + 5)^{-\frac{1}{2}}
\]

Now, let's apply the chain rule. The derivative of \( (8x + 5)^{-\frac{1}{2}} \) with respect to \(x\) is:

\[
\frac{d}{dx} (8x + 5)^{-\frac{1}{2}} = -\frac{1}{2} \cdot (8x + 5)^{-\frac{3}{2}} \cdot 8
\]

### Step 3: Calculate the derivative

Now, taking the derivative of the entire function \(f(x)\):

\[
f'(x) = -4 \cdot \left( -\frac{1}{2} \cdot (8x + 5)^{-\frac{3}{2}} \cdot 8 \right)
\]

Simplifying:

\[
f'(x) = -4 \cdot \left( -4 \cdot (8x + 5)^{-\frac{3}{2}} \right)
\]

\[
f'(x) = 16 \cdot (8x + 5)^{-\frac{3}{2}}
\]

### Final Answer:

\[
f'(x) = \frac{16}{(8x + 5)^{\frac{3}{2}}}
\]

Would you like further clarification or details on any step?

### Related Questions:
1. How does the chain rule work in general for composite functions?
2. What is the derivative of a general power function \( (ax+b)^n \)?
3. Can the quotient rule be used for this function, and how would it compare to the chain rule?
4. How do we simplify fractional exponents when differentiating?
5. What role does the square root play in the derivative of radical functions?

### Tip:
When working with fractional powers like \(\sqrt{x}\), rewriting them as exponents can make applying differentiation rules, like the chain rule, more straightforward!

Find the derivative of f(x) = (-20) / (5 * sqrt(8x + 5)).

Learn how to find the derivative of the function f(x) = (-20) / (5 * sqrt(8x + 5)) step by step. This problem utilizes key calculus concepts such as the chain rule and the power rule for fractional exponents. Suitable for advanced high school or college-level calculus students.

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