For the following exercises, given y=f(u) and u=g(x), find dydx by using Leibniz’s notation for the chain rule: dydx=dydududx.

**214**. 

y=3u−6,u=2x2

215. 

y=6u3,u=7x−4


To find \(\frac{dy}{dx}\) using the chain rule in Leibniz's notation, we need to differentiate \(y\) with respect to \(u\) and \(u\) with respect to \(x\). We'll apply this to both exercises.

### Exercise 214

1. **Given:**
   - \(y = 3u - 6\)
   - \(u = 2x^2\)

2. **Step 1: Find \(\frac{dy}{du}\)**

   \[
   \frac{dy}{du} = 3
   \]

3. **Step 2: Find \(\frac{du}{dx}\)**

   \[
   \frac{du}{dx} = \frac{d}{dx}(2x^2) = 4x
   \]

4. **Step 3: Apply the chain rule:**

   \[
   \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = 3 \cdot 4x = 12x
   \]

### Exercise 215

1. **Given:**
   - \(y = 6u^3\)
   - \(u = 7x - 4\)

2. **Step 1: Find \(\frac{dy}{du}\)**

   \[
   \frac{dy}{du} = 18u^2
   \]

3. **Step 2: Find \(\frac{du}{dx}\)**

   \[
   \frac{du}{dx} = \frac{d}{dx}(7x - 4) = 7
   \]

4. **Step 3: Apply the chain rule:**

   \[
   \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = 18u^2 \cdot 7 = 126u^2
   \]

   Substituting \(u = 7x - 4\):

   \[
   \frac{dy}{dx} = 126(7x - 4)^2
   \]

### Summary of Results
- For Exercise 214: \(\frac{dy}{dx} = 12x\)
- For Exercise 215: \(\frac{dy}{dx} = 126(7x - 4)^2\)

If you need further details or have any questions, let me know! Here are some related questions to consider:

1. How would the derivatives change if \(u\) were a different function of \(x\)?
2. What happens to \(\frac{dy}{dx}\) as \(x\) approaches zero in each case?
3. Can you apply this method to find \(\frac{dy}{dx}\) for trigonometric functions?
4. What is the geometric interpretation of \(\frac{dy}{dx}\)?
5. How does implicit differentiation relate to this chain rule approach?

**Tip:** Always double-check your derivatives for accuracy, especially when substituting values back in!

For the following exercises, given y=f(u) and u=g(x), find dy/dx by using Leibniz’s notation for the chain rule: dy/dx = dy/du * du/dx.

214. y = 3u - 6, u = 2x^2
215. y = 6u^3, u = 7x - 4

Discover how to apply the chain rule to find dy/dx for functions y=3u−6 with u=2x^2, and y=6u^3 with u=7x−4. This solution involves differentiating functions using Leibniz's notation for the chain rule, ideal for students in Grades 11-12.

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