https://r2.erweima.ai/mathbot/image/1726848039212wzlue38k.jpg

Let's analyze the math problem presented in the image step by step. The function \( f(x) \) is defined on the interval \([-2; 0]\) and is given by:

\[
f(x) = (x^2 + 2x + 1)e^{3x + 4}
\]

### Part a) 
**Show that for any real \( x \) in the interval \([-2; 0]\), the derivative \( f'(x) \) is given by**:
\[
f'(x) = (3x^2 + 8x + 5)e^{3x + 4}
\]

We can solve this using the product rule. The function is of the form:

\[
f(x) = u(x) \cdot v(x)
\]
where:
- \( u(x) = x^2 + 2x + 1 \)
- \( v(x) = e^{3x+4} \)

#### Steps:
1. Find \( u'(x) \) and \( v'(x) \):
   - \( u'(x) = 2x + 2 \)
   - \( v'(x) = 3e^{3x+4} \) (from the chain rule)

2. Use the product rule for differentiation:
\[
f'(x) = u'(x)v(x) + u(x)v'(x)
\]
Substitute the expressions for \( u(x), u'(x), v(x), \) and \( v'(x) \):
\[
f'(x) = (2x + 2)e^{3x+4} + (x^2 + 2x + 1)3e^{3x+4}
\]

Factor out \( e^{3x+4} \):
\[
f'(x) = e^{3x+4} \left( (2x + 2) + 3(x^2 + 2x + 1) \right)
\]

Simplify the terms inside the parentheses:
\[
(2x + 2) + 3(x^2 + 2x + 1) = 2x + 2 + 3x^2 + 6x + 3 = 3x^2 + 8x + 5
\]

Thus, we have:
\[
f'(x) = (3x^2 + 8x + 5)e^{3x+4}
\]

This completes part **(a)**.

---

Would you like to proceed with parts **(b)**, **(c)**, and **(d)**, or have any further questions on this part?

Here are 5 relative questions for further exploration:
1. How do you apply the product rule in differentiation?
2. What is the chain rule and how is it used in derivative computations?
3. How can you verify that the derivative is correctly simplified?
4. What is the significance of factoring out exponential terms in derivatives?
5. How do derivatives affect the shape and behavior of a function?

**Tip**: When differentiating composite functions, always identify the inner and outer functions to apply the chain rule correctly.

Show that for any real x in the interval [-2; 0], the derivative f'(x) is given by f'(x) = (3x^2 + 8x + 5)e^{3x + 4}

Explore the detailed step-by-step differentiation of the function (x^2 + 2x + 1)e^{3x + 4} within the interval [-2, 0]. This guide applies the product and chain rule, crucial concepts in calculus, suitable for university-level or advanced placement courses.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation