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The problem asks you to use the definition of the derivative to find \( f'(x) \) for the function \( f(x) = \frac{27}{x} \), and then evaluate the derivative at specific points.

### Step 1: Definition of the Derivative

The definition of the derivative is given by:

\[
f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}
\]

### Step 2: Substitute the Function into the Definition

Substituting \( f(x) = \frac{27}{x} \) into the definition:

\[
f'(x) = \lim_{h \to 0} \frac{\frac{27}{x + h} - \frac{27}{x}}{h}
\]

### Step 3: Simplify the Expression Inside the Limit

To simplify, find a common denominator for the fractions inside the limit:

\[
f'(x) = \lim_{h \to 0} \frac{\frac{27x - 27(x + h)}{x(x + h)}}{h} = \lim_{h \to 0} \frac{27x - 27x - 27h}{h \cdot x(x + h)}
\]

\[
f'(x) = \lim_{h \to 0} \frac{-27h}{h \cdot x(x + h)}
\]

Simplify further:

\[
f'(x) = \lim_{h \to 0} \frac{-27}{x(x + h)}
\]

As \( h \to 0 \):

\[
f'(x) = \frac{-27}{x^2}
\]

### Step 4: Evaluate the Derivative at Specific Points

1. \( f'(-3) = \frac{-27}{(-3)^2} = \frac{-27}{9} = -3 \)
2. \( f'(0) \) does not exist because division by zero is undefined.
3. \( f'(4) = \frac{-27}{4^2} = \frac{-27}{16} \)

Would you like further details on any of these steps or have any specific questions? Here are some follow-up questions to consider:

1. How do you determine when a derivative does not exist?
2. Can you apply the derivative formula to other types of functions?
3. What happens to the derivative when the function is undefined at a certain point?
4. How does the derivative behave as the value of \(x\) increases or decreases?
5. What are some common mistakes when calculating derivatives using limits?

**Tip:** Always check the domain of the function before evaluating the derivative at specific points, as some points may lead to undefined expressions!

Using the definition of the derivative, find f'(x). Then find f'(-3), f'(0), and f'(4) when the derivative exists for f(x) = 27/x.

Find the derivative of the function f(x) = 27/x using the definition of the derivative. Learn how to compute f'(-3), f'(0), and f'(4) by applying limits and simplifying the expression. Suitable for calculus students at the high school or college level.

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