https://file.aiquickdraw.com/mathbot/file/1726020281160sa9b2upx.jpg

The function given is:

\[
f(x) = 14x^2 + 11x - 7
\]

We need to:

1. Find the derivative \( f'(x) \).
2. Evaluate the derivative at \( x = 12 \) to get \( f'(12) \).

### Step 1: Find the derivative

Using basic rules of differentiation:
- The derivative of \( 14x^2 \) is \( 28x \) (power rule).
- The derivative of \( 11x \) is \( 11 \) (since the derivative of \( x \) is 1).
- The derivative of \( -7 \) is 0 (since the derivative of a constant is 0).

So, the derivative is:

\[
f'(x) = 28x + 11
\]

### Step 2: Evaluate the derivative at \( x = 12 \)

Substitute \( x = 12 \) into \( f'(x) \):

\[
f'(12) = 28(12) + 11 = 336 + 11 = 347
\]

Thus, the value of \( f'(12) \) is:

\[
f'(12) = 347
\]

Would you like more details or have any questions?

Here are 5 related questions to explore:
1. How do we differentiate polynomial functions in general?
2. What is the power rule in differentiation?
3. How does differentiation apply to real-life problems?
4. Can we find the second derivative of \( f(x) \) and interpret its meaning?
5. How do we handle more complex functions involving product or chain rules?

**Tip:** Always apply the power rule carefully, ensuring that you reduce the power of \( x \) by 1 and multiply by the original exponent.

Find the derivative of the function f(x) = 14x^2 + 11x - 7 at number 12.

Learn how to find the derivative of the polynomial function f(x) = 14x^2 + 11x - 7 and evaluate it at x = 12. This example covers differentiation using the power rule and is suitable for students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation