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We are asked to find \(\frac{dy}{dx}\) given:

\[
y = 9u + 7 \quad \text{and} \quad u = 7x - 8.
\]

We will use the chain rule to solve this, since \(y\) is a function of \(u\), and \(u\) is a function of \(x\). The chain rule states:

\[
\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}.
\]

### Step 1: Differentiate \(y\) with respect to \(u\)
Given \(y = 9u + 7\), the derivative of \(y\) with respect to \(u\) is:

\[
\frac{dy}{du} = 9.
\]

### Step 2: Differentiate \(u\) with respect to \(x\)
Given \(u = 7x - 8\), the derivative of \(u\) with respect to \(x\) is:

\[
\frac{du}{dx} = 7.
\]

### Step 3: Apply the chain rule
Now, applying the chain rule:

\[
\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = 9 \cdot 7 = 63.
\]

Thus, \(\frac{dy}{dx} = 63\).

Would you like more details or have any questions? Here are some related questions for deeper understanding:

1. What is the chain rule, and how does it work in differentiation?
2. How does the chain rule differ from the product and quotient rules?
3. What would happen if \(y\) were a more complex function of \(u\)?
4. How does implicit differentiation work in problems like this?
5. Could we solve this problem using implicit differentiation?

**Tip:** In problems involving functions of multiple variables, always check if the chain rule applies to simplify the solution.

Find dy/dx, if y = 9u + 7 and u = 7x - 8.

Learn how to find dy/dx by applying the chain rule, given that y = 9u + 7 and u = 7x - 8. This problem demonstrates the use of basic calculus concepts and is suitable for students in Grades 11-12.

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