Given  
f
(
x
)
=
3
x
−
2
 , find  
f
′
(
1
)
  using the definition of a derivative.

To find \( f'(1) \) using the definition of a derivative, we use the limit definition of the derivative:

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

In this case, \( f(x) = 3x - 2 \), and we are asked to find \( f'(1) \). So, we set \( a = 1 \) and proceed step by step.

### Step 1: Substitute \( f(x) = 3x - 2 \) and \( a = 1 \) into the limit definition:
\[
f'(1) = \lim_{h \to 0} \frac{f(1+h) - f(1)}{h}
\]

### Step 2: Compute \( f(1+h) \) and \( f(1) \)
- \( f(1+h) = 3(1+h) - 2 = 3 + 3h - 2 = 1 + 3h \)
- \( f(1) = 3(1) - 2 = 1 \)

### Step 3: Substitute these into the limit expression:
\[
f'(1) = \lim_{h \to 0} \frac{(1 + 3h) - 1}{h}
\]

### Step 4: Simplify the expression:
\[
f'(1) = \lim_{h \to 0} \frac{1 + 3h - 1}{h} = \lim_{h \to 0} \frac{3h}{h}
\]

### Step 5: Cancel \( h \) and compute the limit:
\[
f'(1) = \lim_{h \to 0} 3 = 3
\]

### Final Answer:
\[
f'(1) = 3
\]

Would you like more details, or do you have any questions?

Here are 5 related questions:
1. How would you find \( f'(2) \) using the same method?
2. What is the geometric meaning of the derivative at \( x = 1 \)?
3. How does the derivative of \( f(x) = 3x - 2 \) compare to the slope of the line?
4. Can you apply the definition of the derivative to a nonlinear function like \( f(x) = x^2 \)?
5. What happens to the derivative if the function is constant?

**Tip:** The derivative of a linear function like \( f(x) = 3x - 2 \) is always the slope of the line, which remains constant for all \( x \).

Given f(x) = 3x − 2, find f′(1) using the definition of a derivative.

Learn how to find the derivative of f(x) = 3x − 2 at x = 1 using the limit definition of a derivative. This step-by-step solution involves key calculus concepts such as the limit process and the definition of a derivative. Suitable for Grades 11-12 students.

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