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Use the chain rule formula to find the derivative of the following functions. Express your final answers in terms of the variable x

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!

Use the chain rule formula to find the derivative of the following functions: 3. y = (u - 4)^(5/2) where u = x^4 + 4, 4. y = 3u - 2 where u = 3x, 5. y = u^2 + 3u - 7 where u = 2x + 1. Express your final answers in terms of the variable x.

Learn how to apply the chain rule to differentiate composite functions. This guide covers three problems: (1) y = (u - 4)^(5/2) where u = x^4 + 4, (2) y = 3u - 2 where u = 3x, and (3) y = u^2 + 3u - 7 where u = 2x + 1. Step-by-step solutions are provided with all final answers in terms of x. Suitable for advanced high school students studying calculus.

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