Math Problem Statement

its x^2 not x^3

Solution

Thank you for the clarification. Let's go through the steps again with the correct function f(x)=4xx2+2f(x) = 4x - x^2 + 2.

Step 1: Find the derivative of f(x)f(x).

The function is: f(x)=4xx2+2f(x) = 4x - x^2 + 2 The derivative f(x)f'(x) with respect to xx is: f(x)=ddx(4x)ddx(x2)+ddx(2)f'(x) = \frac{d}{dx}(4x) - \frac{d}{dx}(x^2) + \frac{d}{dx}(2)

Let's differentiate each term:

  • ddx(4x)=4\frac{d}{dx}(4x) = 4
  • ddx(x2)=2x\frac{d}{dx}(x^2) = 2x
  • ddx(2)=0\frac{d}{dx}(2) = 0 (since the derivative of a constant is zero)

So, the derivative f(x)f'(x) is: f(x)=42xf'(x) = 4 - 2x

Step 2: Evaluate the derivative at x=2x = -2.

Substitute x=2x = -2 into f(x)f'(x): f(2)=42(2)f'(-2) = 4 - 2(-2) f(2)=4+4=8f'(-2) = 4 + 4 = 8

Final Answer:

The rate of change of the function at x=2x = -2 is 88.

Would you like more details or have any questions?

Here are five related questions you might find useful:

  1. How does changing the coefficients in a quadratic function affect its derivative?
  2. What is the geometric interpretation of the rate of change for a quadratic function?
  3. How do you determine the maximum or minimum points using the derivative?
  4. How can you apply the concept of derivatives to real-world problems, such as velocity?
  5. What is the significance of the sign of the derivative in different intervals of the function?

Tip: Always ensure that the correct form of the function is used before performing calculations to avoid errors in the final result.

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Math Problem Analysis

Mathematical Concepts

Differentiation
Quadratic Functions

Formulas

Derivative rules

Theorems

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Suitable Grade Level

Grades 11-12