https://file.aiquickdraw.com/mathbot/file/1722583236489vknze79l.jpg

To differentiate the function \( f(x) = 2x^2 + 3x - 5 \), we can apply basic differentiation rules. Here are the steps:

1. **Identify the function components**: The function \( f(x) \) consists of three terms:
   - \( 2x^2 \)
   - \( 3x \)
   - \(-5\)

2. **Differentiate each term separately**:
   - For the term \( 2x^2 \): Use the power rule, which states that \(\frac{d}{dx}(x^n) = nx^{n-1}\).
     \[
     \frac{d}{dx}(2x^2) = 2 \cdot 2x^{2-1} = 4x
     \]
   - For the term \( 3x \): Use the power rule, where \( x \) is \( x^1 \).
     \[
     \frac{d}{dx}(3x) = 3 \cdot 1x^{1-1} = 3
     \]
   - For the constant term \(-5\): The derivative of a constant is zero.
     \[
     \frac{d}{dx}(-5) = 0
     \]

3. **Combine the differentiated terms**: Add the derivatives of each term to get the derivative of the entire function.
   \[
   f'(x) = 4x + 3
   \]

So, the derivative of \( f(x) = 2x^2 + 3x - 5 \) is \( f'(x) = 4x + 3 \).

Would you like more details or have any questions?

Here are 8 related questions for further exploration:
1. How do you find the second derivative of \( f(x) = 2x^2 + 3x - 5 \)?
2. What is the derivative of the function \( g(x) = x^3 - 4x^2 + 7x - 10 \)?
3. How do you apply the chain rule to the function \( h(x) = (3x^2 + 2x - 1)^5 \)?
4. What are the critical points of the function \( f(x) = 2x^2 + 3x - 5 \)?
5. How can you use differentiation to find the local maxima and minima of a function?
6. What is the derivative of the exponential function \( e^x \)?
7. How do you differentiate the trigonometric function \( \sin(x) \)?
8. What are the applications of differentiation in real-world problems?

**Tip:** When differentiating polynomial functions, remember to apply the power rule term-by-term for accurate results.

Explore the differentiation rule with a step-by-step explanation of how to differentiate the function f(x) = 2x^2 + 3x - 5. Learn the power rule and constant rule applications in differentiation. Suitable for advanced high school students and those preparing for calculus.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation