https://file.aiquickdraw.com/mathbot/file/1726805866956oyu71tt0.jpg

Let's go through the problems shown in the image:

### Problem 2:
We need to find the derivatives using the **Product Rule** and then evaluate the specific derivatives.

#### 1. Given:
\[
F(x) = x^4e^x
\]
The product rule states:
\[
(f(x)g(x))' = f'(x)g(x) + f(x)g'(x)
\]
Here, \( f(x) = x^4 \) and \( g(x) = e^x \).

##### Derivative:
- \( f'(x) = 4x^3 \)
- \( g'(x) = e^x \)

Using the product rule:
\[
F'(x) = 4x^3 e^x + x^4 e^x
\]

##### Evaluating \( F'(1) \):
Substitute \( x = 1 \):
\[
F'(1) = 4(1)^3 e^1 + (1)^4 e^1 = 4e + e = 5e
\]

#### 2. Given:
\[
G(x) = \left(\frac{x^9}{x^{1/3}}\right)2^x
\]
First, simplify the expression inside the parentheses:
\[
G(x) = x^{9 - 1/3} 2^x = x^{26/3} 2^x
\]

Again, using the product rule:
Let \( f(x) = x^{26/3} \) and \( g(x) = 2^x \).

##### Derivative:
- \( f'(x) = \frac{26}{3} x^{23/3} \)
- \( g'(x) = 2^x \ln 2 \)

Now, applying the product rule:
\[
G'(x) = \frac{26}{3} x^{23/3} 2^x + x^{26/3} 2^x \ln 2
\]

##### Evaluating \( G'(-1) \):
Substitute \( x = -1 \):
\[
G'(-1) = \frac{26}{3} (-1)^{23/3} 2^{-1} + (-1)^{26/3} 2^{-1} \ln 2
\]
Note that powers of negative numbers like \( (-1)^{23/3} \) can be complex, but the calculation could simplify depending on how these powers are handled.

#### 3. Given:
\[
z = \frac{\pi^2}{x^3}
\]
We want to compute:
\[
\frac{dz}{dx} \bigg|_{x=1}
\]

First, take the derivative using the power rule:
\[
\frac{dz}{dx} = -3 \frac{\pi^2}{x^4}
\]

Now, substitute \( x = 1 \):
\[
\frac{dz}{dx} \bigg|_{x=1} = -3 \frac{\pi^2}{1^4} = -3\pi^2
\]

---

### Problem 3:
We are asked to compute the derivative of the product \( f(x)g(x) \) in two ways:

#### Given:
\( f(x) = 2x^6 \) and \( g(x) = 4x^2 + x \).

1. **Using the product rule**:
\[
(f(x)g(x))' = f'(x)g(x) + f(x)g'(x)
\]

- \( f'(x) = 12x^5 \)
- \( g'(x) = 8x + 1 \)

So:
\[
(f(x)g(x))' = 12x^5(4x^2 + x) + 2x^6(8x + 1)
\]

Simplify each term:
\[
= 12x^5(4x^2 + x) = 48x^7 + 12x^6
\]
\[
= 2x^6(8x + 1) = 16x^7 + 2x^6
\]

Adding these:
\[
(f(x)g(x))' = 48x^7 + 12x^6 + 16x^7 + 2x^6 = 64x^7 + 14x^6
\]

2. **Distribute the product first**:
First, expand \( f(x)g(x) \):
\[
f(x)g(x) = 2x^6(4x^2 + x) = 8x^8 + 2x^7
\]

Now, take the derivative of the expanded form:
\[
\frac{d}{dx} \left( 8x^8 + 2x^7 \right) = 64x^7 + 14x^6
\]

Both methods give the same result, as expected.

---

Do you have any questions or would you like further details on any of these problems?

### Related questions:
1. How would the product rule apply if you had three functions multiplied together?
2. Can you apply the quotient rule for the function in part 2 of Problem 2?
3. How does the derivative of exponential functions like \( e^x \) compare to \( 2^x \)?
4. What happens to the derivative if you take higher-order derivatives of the functions in Problem 3?
5. Can the complex result from \( G'(-1) \) in Problem 2 be simplified in any meaningful way?

### Tip:
When dealing with powers of negative numbers in fractional form (like \( x^{23/3} \)), carefully consider if the result might involve complex numbers.

Find the derivative of the given function using the product rule, apply it to specific values, and distribute or simplify as needed.

Solve derivative problems using the product rule and apply to exponential and power functions. Problems include simplifying algebraic expressions and calculating specific derivative values. Suitable for high school or college calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation