Let's go through each problem step by step, using the chain rule to differentiate the given functions. We will express all the final answers in terms of $x$.

Problem 3:

$y = (u - 4)^{\frac{5}{2}} \quad \text{where} \quad u = x^4 + 4$

To differentiate $y$ with respect to $x$, we need to apply the chain rule.

First, differentiate $y$ with respect to $u$: $\frac{dy}{du} = \frac{5}{2} (u - 4)^{\frac{3}{2}}$

Next, differentiate $u$ with respect to $x$: $\frac{du}{dx} = 4x^3$

Now, apply the chain rule: $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = \frac{5}{2} (u - 4)^{\frac{3}{2}} \cdot 4x^3$

Substitute $u = x^4 + 4$ into the expression: $\frac{dy}{dx} = \frac{5}{2} \left( (x^4 + 4) - 4 \right)^{\frac{3}{2}} \cdot 4x^3$ $\frac{dy}{dx} = 10x^3 (x^4)^{\frac{3}{2}} = 10x^3 \cdot x^6 = 10x^9$

Problem 4:

$y = 3u - 2 \quad \text{where} \quad u = 3x$

First, differentiate $y$ with respect to $u$: $\frac{dy}{du} = 3$

Next, differentiate $u$ with respect to $x$: $\frac{du}{dx} = 3$

Now, apply the chain rule: $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = 3 \cdot 3 = 9$

Problem 5:

$y = u^2 + 3u - 7 \quad \text{where} \quad u = 2x + 1$

First, differentiate $y$ with respect to $u$: $\frac{dy}{du} = 2u + 3$

Next, differentiate $u$ with respect to $x$: $\frac{du}{dx} = 2$

Now, apply the chain rule: $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = (2u + 3) \cdot 2$ Substitute $u = 2x + 1$: $\frac{dy}{dx} = (2(2x + 1) + 3) \cdot 2 = (4x + 2 + 3) \cdot 2 = (4x + 5) \cdot 2$ $\frac{dy}{dx} = 8x + 10$

Do you need further clarifications on any of the steps?

Here are 5 related questions:

1. How does the chain rule apply to multi-variable functions?
2. What are some real-life applications of the chain rule?
3. Can you explain why we differentiate $y$ with respect to $u$ first in these problems?
4. What would happen if $u$ was a more complex function of $x$?
5. How does the chain rule change when applied to implicit differentiation?

Tip: Always start by identifying the outer and inner functions when using the chain rule!