The problem is asking us to use the chain rule to compute the derivative $\frac{dy}{dx}$ for the given functions.

Problem:

1. $y = u^2 + 1$
2. $u = 3x - 2$

Solution:

Using the chain rule, we know: $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}.$

Step 1: Compute $\frac{dy}{du}$

From $y = u^2 + 1$: $\frac{dy}{du} = 2u.$

Step 2: Compute $\frac{du}{dx}$

From $u = 3x - 2$: $\frac{du}{dx} = 3.$

Step 3: Combine using the chain rule

$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = (2u) \cdot 3 = 6u.$

Step 4: Substitute $u = 3x - 2$

$\frac{dy}{dx} = 6(3x - 2) = 18x - 12.$

Final Answer:

$\frac{dy}{dx} = 18x - 12.$

Would you like to see further examples or details on using the chain rule?

Related Questions:

1. How does the chain rule apply to functions with more than two nested layers?
2. What are some real-world applications of derivatives calculated using the chain rule?
3. How does the chain rule work for trigonometric functions?
4. Can we compute higher-order derivatives using the chain rule?
5. What is the difference between the chain rule and product rule in differentiation?

Tip:

Always keep track of each substitution when applying the chain rule, especially for nested functions, to avoid errors.