https://file.aiquickdraw.com/mathbot/file/17255781494449lgg3lp2.jpg

It seems that the image's text couldn't be extracted properly. However, I can attempt to read the mathematical content visually and solve it for you.

The image shows a problem where you are asked to "Find the derivative of \( f(x) = 4x^3 + 2x^2 \) using the limit definition of a derivative."

The limit definition of the derivative is:

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

Let’s solve this step by step:

### Step 1: Find \( f(x+h) \)

Given \( f(x) = 4x^3 + 2x^2 \), calculate \( f(x+h) \):

\[
f(x+h) = 4(x+h)^3 + 2(x+h)^2
\]

Expand both terms:

\[
4(x+h)^3 = 4(x^3 + 3x^2h + 3xh^2 + h^3) = 4x^3 + 12x^2h + 12xh^2 + 4h^3
\]
\[
2(x+h)^2 = 2(x^2 + 2xh + h^2) = 2x^2 + 4xh + 2h^2
\]

So, \( f(x+h) \) becomes:

\[
f(x+h) = (4x^3 + 2x^2) + (12x^2h + 4xh) + (12xh^2 + 2h^2 + 4h^3)
\]

### Step 2: Apply the limit definition

Now substitute \( f(x+h) \) and \( f(x) \) into the limit definition:

\[
f'(x) = \lim_{{h \to 0}} \frac{{(4x^3 + 12x^2h + 12xh^2 + 4h^3 + 2x^2 + 4xh + 2h^2) - (4x^3 + 2x^2)}}{h}
\]

Simplifying the expression:

\[
f'(x) = \lim_{{h \to 0}} \frac{{12x^2h + 4xh + 12xh^2 + 2h^2 + 4h^3}}{h}
\]

Factor out \( h \):

\[
f'(x) = \lim_{{h \to 0}} \frac{{h(12x^2 + 4x + 12xh + 2h + 4h^2)}}{h}
\]

Cancel \( h \) in the numerator and denominator:

\[
f'(x) = \lim_{{h \to 0}} (12x^2 + 4x + 12xh + 2h + 4h^2)
\]

### Step 3: Take the limit as \( h \to 0 \)

As \( h \to 0 \), the terms involving \( h \) vanish:

\[
f'(x) = 12x^2 + 4x
\]

So, the derivative of \( f(x) = 4x^3 + 2x^2 \) is:

\[
f'(x) = 12x^2 + 4x
\]

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

Here are some related questions for further exploration:
1. How would you differentiate \( f(x) = 4x^3 + 2x^2 \) using standard rules of differentiation?
2. Can you apply the same limit definition to find the derivative of \( f(x) = x^2 \)?
3. What happens if we take the second derivative of \( f(x) = 4x^3 + 2x^2 \)?
4. Can you find the derivative of a polynomial function using both methods: the limit definition and power rule?
5. How does the limit definition help in understanding the concept of instantaneous rate of change?

**Tip**: Using the limit definition helps reinforce your understanding of what the derivative truly represents — the slope of the tangent line at a point.

Find the derivative of f(x) = 4x^3 + 2x^2 using the limit definition of a derivative.

Solve the derivative of f(x) = 4x^3 + 2x^2 using the limit definition of a derivative with detailed step-by-step explanations. This problem covers calculus concepts including limits and the derivative of a polynomial function. Suitable for Grades 11-12 and 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