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.