create me a random equation in latex about derivative and explain how u got the answer

Here's a random equation involving derivatives, written in LaTeX:

\[
f'(x) = \frac{d}{dx} \left( x^3 + 2x^2 - 5x + 7 \right) = 3x^2 + 4x - 5
\]

### Explanation

1. **Original Function**: \( f(x) = x^3 + 2x^2 - 5x + 7 \)

   This is a polynomial function where the highest degree is 3 (cubic polynomial).

2. **Taking the Derivative**: 
   - The derivative of \( x^n \) with respect to \( x \) is \( n \cdot x^{n-1} \).
   - Applying this rule:
     - \( \frac{d}{dx} \left( x^3 \right) = 3x^2 \)
     - \( \frac{d}{dx} \left( 2x^2 \right) = 4x \)
     - \( \frac{d}{dx} \left( -5x \right) = -5 \)
     - The derivative of the constant \( 7 \) is \( 0 \).

3. **Final Answer**: 
   - Combine these results to get the derivative:
     \[
     f'(x) = 3x^2 + 4x - 5
     \]

Would you like more details or have any questions?

### Related Questions:
1. How do you find the derivative of trigonometric functions?
2. What is the geometric interpretation of a derivative?
3. How do you calculate higher-order derivatives?
4. What is the chain rule in differentiation?
5. How does the derivative apply to real-world problems?

**Tip**: When differentiating, always simplify the function first if possible; this can make the differentiation process easier.

Explore a random equation involving derivatives in LaTeX with a detailed explanation. Learn how to apply the power rule of differentiation to a polynomial function. Suitable for Grades 10-12 students and anyone interested in calculus basics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation