Let f(x) = 3x
x + 2 . Use the limit definition of the
derivative to find f′(1). No credit will be given for
methods not using the limit definition of the
derivative.

To find \( f'(1) \) using the limit definition of the derivative, we follow this approach:

The limit definition of the derivative is given by:
\[
f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}
\]
We are asked to find \( f'(1) \), so we substitute \( x = 1 \) into this formula:
\[
f'(1) = \lim_{h \to 0} \frac{f(1+h) - f(1)}{h}
\]

### Step 1: Find \( f(1) \)
We are given that \( f(x) = \frac{3x}{x + 2} \). To find \( f(1) \):
\[
f(1) = \frac{3(1)}{1 + 2} = \frac{3}{3} = 1
\]

### Step 2: Find \( f(1+h) \)
Now, we calculate \( f(1+h) \):
\[
f(1+h) = \frac{3(1+h)}{(1+h) + 2} = \frac{3(1+h)}{h+3} = \frac{3 + 3h}{h+3}
\]

### Step 3: Set up the limit
We substitute these values into the limit definition of the derivative:
\[
f'(1) = \lim_{h \to 0} \frac{f(1+h) - f(1)}{h} = \lim_{h \to 0} \frac{\frac{3 + 3h}{h+3} - 1}{h}
\]
Simplify the expression inside the limit:
\[
f'(1) = \lim_{h \to 0} \frac{\frac{3 + 3h}{h+3} - \frac{h+3}{h+3}}{h}
\]
\[
f'(1) = \lim_{h \to 0} \frac{\frac{(3 + 3h) - (h+3)}{h+3}}{h}
\]
Simplify the numerator:
\[
f'(1) = \lim_{h \to 0} \frac{\frac{3 + 3h - h - 3}{h+3}}{h} = \lim_{h \to 0} \frac{\frac{2h}{h+3}}{h}
\]

### Step 4: Simplify the limit
Now, simplify further:
\[
f'(1) = \lim_{h \to 0} \frac{2h}{h(h+3)} = \lim_{h \to 0} \frac{2}{h+3}
\]
As \( h \to 0 \), we substitute \( h = 0 \) in the expression:
\[
f'(1) = \frac{2}{3}
\]

### Final Answer:
\[
f'(1) = \frac{2}{3}
\]

Let me know if you'd like further details or have any questions!

Here are 5 related questions to expand on this topic:
1. How would the derivative change if \( f(x) = \frac{5x}{x + 3} \)?
2. What is the geometric interpretation of \( f'(x) \)?
3. How can the chain rule be applied to differentiate more complex functions?
4. What is the difference between the limit definition of a derivative and other differentiation rules?
5. How does the derivative of a rational function behave near vertical asymptotes?

**Tip**: The limit definition of the derivative helps build a strong understanding of the concept of instantaneous rate of change. It is fundamental for calculus!

Let f(x) = 3x / (x + 2). Use the limit definition of the derivative to find f′(1). No credit will be given for methods not using the limit definition of the derivative.

Learn how to find the derivative of the function f(x) = 3x / (x + 2) at x = 1 using the limit definition of the derivative. This detailed step-by-step solution explains key calculus concepts like limits and rational functions, suitable for students in advanced high school or introductory calculus.

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