Let 
f(x) = 
4x
5x2 + 9
.
 Compute the difference quotient for f(x) at x = 7 with h = 0.2.

To compute the difference quotient for \( f(x) = \frac{4x}{5x^2 + 9} \) at \( x = 7 \) with \( h = 0.2 \), we use the formula for the difference quotient:

\[
\text{Difference Quotient} = \frac{f(x+h) - f(x)}{h}
\]

Given \( f(x) = \frac{4x}{5x^2 + 9} \), \( x = 7 \), and \( h = 0.2 \), we need to calculate \( f(7) \) and \( f(7 + 0.2) = f(7.2) \).

### Step 1: Compute \( f(7) \)
\[
f(7) = \frac{4(7)}{5(7)^2 + 9} = \frac{28}{5(49) + 9} = \frac{28}{245 + 9} = \frac{28}{254}
\]

### Step 2: Compute \( f(7.2) \)
\[
f(7.2) = \frac{4(7.2)}{5(7.2)^2 + 9} = \frac{28.8}{5(51.84) + 9} = \frac{28.8}{259.2 + 9} = \frac{28.8}{268.2}
\]

### Step 3: Compute the difference quotient
Now that we have \( f(7) = \frac{28}{254} \) and \( f(7.2) = \frac{28.8}{268.2} \), we can compute the difference quotient:

\[
\text{Difference Quotient} = \frac{f(7.2) - f(7)}{0.2} = \frac{\frac{28.8}{268.2} - \frac{28}{254}}{0.2}
\]

This requires simplifying the expression. Let me calculate this next.The difference quotient for \( f(x) = \frac{4x}{5x^2 + 9} \) at \( x = 7 \) with \( h = 0.2 \) is approximately \( -0.01427 \).

Would you like more details on how this result was computed or have any additional questions?

Here are some related questions for further exploration:

1. How is the difference quotient related to the derivative?
2. What happens to the difference quotient as \( h \) approaches 0?
3. Can you compute the derivative of \( f(x) = \frac{4x}{5x^2 + 9} \) using the limit definition of a derivative?
4. What is the physical interpretation of the difference quotient in real-world scenarios?
5. How does changing \( h \) affect the accuracy of the approximation for the derivative?

**Tip:** The difference quotient is an approximation of the derivative when \( h \) is small. The smaller \( h \) becomes, the closer the difference quotient gets to the actual derivative.

Let f(x) = 4x / (5x^2 + 9). Compute the difference quotient for f(x) at x = 7 with h = 0.2.

Learn how to compute the difference quotient for the rational function f(x) = 4x / (5x^2 + 9) at x = 7 with h = 0.2. This step-by-step solution breaks down the process, covering key algebraic concepts and the difference quotient formula. Suitable for high school students in Grades 10-12.

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