Find the derivative of the function using the definition of derivative.
f(x) =
1

9 + x
f '(x) =

The given function is:

\[
f(x) = \frac{1}{9 + x}
\]

We are tasked with finding the derivative \( f'(x) \) using the definition of the derivative. The definition of the derivative is:

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

Let's apply this step by step.

### Step 1: Write the expression for \( f(x+h) \)
The function is \( f(x) = \frac{1}{9 + x} \), so:

\[
f(x+h) = \frac{1}{9 + (x+h)} = \frac{1}{9 + x + h}
\]

### Step 2: Set up the difference quotient
Substitute \( f(x+h) \) and \( f(x) \) into the difference quotient:

\[
\frac{f(x+h) - f(x)}{h} = \frac{\frac{1}{9 + x + h} - \frac{1}{9 + x}}{h}
\]

### Step 3: Combine the fractions in the numerator
To subtract the two fractions, we find a common denominator:

\[
\frac{1}{9 + x + h} - \frac{1}{9 + x} = \frac{(9 + x) - (9 + x + h)}{(9 + x + h)(9 + x)} = \frac{-(h)}{(9 + x + h)(9 + x)}
\]

Thus, the difference quotient becomes:

\[
\frac{\frac{-(h)}{(9 + x + h)(9 + x)}}{h}
\]

### Step 4: Simplify the expression
Simplify the difference quotient:

\[
= \frac{-h}{h(9 + x + h)(9 + x)} = \frac{-1}{(9 + x + h)(9 + x)}
\]

### Step 5: Take the limit as \( h \to 0 \)
Now, we take the limit of the expression as \( h \to 0 \):

\[
f'(x) = \lim_{h \to 0} \frac{-1}{(9 + x + h)(9 + x)} = \frac{-1}{(9 + x)^2}
\]

### Final Answer:
The derivative of the function is:

\[
f'(x) = \frac{-1}{(9 + x)^2}
\]

Would you like more details or further clarification?

### Related Questions:
1. What is the geometric interpretation of the derivative at a point?
2. How would the derivative change if the function were \( \frac{1}{9 - x} \)?
3. Can you apply the quotient rule to find the derivative of \( f(x) = \frac{1}{9 + x} \)?
4. How does the derivative relate to the rate of change for this function?
5. What would the second derivative of \( f(x) \) be?

### Tip:
Using the definition of the derivative is especially useful for understanding the concept of limits and how the slope of the tangent line is derived.

Find the derivative of the function using the definition of derivative. f(x) = 1 / (9 + x)

This page explains how to find the derivative of the function f(x) = 1 / (9 + x) using the definition of the derivative. Follow a detailed step-by-step process that covers the difference quotient, limit calculation, and final result. Suitable for advanced high school or college calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation