https://file.aiquickdraw.com/mathbot/file/1726449884307jjmd6h1r.jpg

To solve this problem, we will use the four-step definition of the derivative, which is:

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

Given that \( f(x) = 6x + 8 \), we can now follow the four steps.

### Step 1: Compute \( f(x+h) \)
Substitute \( x+h \) into the given function \( f(x) = 6x + 8 \):

\[
f(x+h) = 6(x+h) + 8 = 6x + 6h + 8
\]

### Step 2: Compute \( f(x+h) - f(x) \)
Now subtract \( f(x) \) from \( f(x+h) \):

\[
f(x+h) - f(x) = (6x + 6h + 8) - (6x + 8) = 6h
\]

### Step 3: Divide by \( h \)
Next, divide the result from step 2 by \( h \):

\[
\frac{f(x+h) - f(x)}{h} = \frac{6h}{h} = 6
\]

### Step 4: Take the limit as \( h \to 0 \)
Finally, take the limit of the expression as \( h \to 0 \):

\[
\lim_{h \to 0} 6 = 6
\]

Thus, the derivative \( f'(x) \) is \( 6 \).

### Filling in the Boxes
1. \( f(x+h) = 6x + 6h + 8 \)
2. \( f(x+h) - f(x) = 6h \)
3. \( \frac{f(x+h) - f(x)}{h} = 6 \)
4. \( \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} = 6 \)

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

Here are 5 related questions to expand your understanding:
1. How would the derivative change if \( f(x) = 6x^2 + 8 \)?
2. What does the derivative tell us about the slope of the function \( f(x) = 6x + 8 \)?
3. Can you apply the same process to find the derivative of \( f(x) = x^3 + 4x^2 \)?
4. How does the concept of the derivative relate to rates of change in real-world scenarios?
5. How would you use the power rule to directly find the derivative of \( f(x) = 6x + 8 \)?

**Tip:** The limit definition of the derivative is fundamental for understanding the concept of instantaneous rate of change and can be used to derive the power rule.

Use the four-step definition of the derivative to find f'(x) if f(x) = 6x + 8.

Step-by-step guide on using the four-step definition of the derivative to calculate f'(x) for the function f(x) = 6x + 8. This example illustrates key calculus concepts including the limit definition of a derivative. Suitable for college students or advanced high school learners in calculus courses.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation