To find the derivative of the function \( y = e^{7-x} \), we use the chain rule. Here's the step-by-step process:

1. **Identify the outer function and inner function**:
   - The outer function is \( e^u \), where \( u = 7 - x \).

2. **Differentiate the outer function**:
   - The derivative of \( e^u \) with respect to \( u \) is \( e^u \).

3. **Differentiate the inner function**:
   - The derivative of \( u = 7 - x \) with respect to \( x \) is \( -1 \).

4. **Apply the chain rule**:
   \[
   \frac{dy}{dx} = e^{7 - x} \cdot (-1)
   \]

So, the derivative is:
\[
\frac{dy}{dx} = -e^{7 - x}
\]

Would you like more details or have any questions?

Here are five related questions to expand your understanding:
1. What is the derivative of \( y = e^{2x - 5} \)?
2. How do you differentiate \( y = e^{x^2} \)?
3. What is the second derivative of \( y = e^{7 - x} \)?
4. How would you differentiate \( y = e^{\sin(x)} \)?
5. How does the chain rule apply when differentiating functions involving logarithms?

**Tip:** The chain rule is essential when differentiating composite functions—always break the function into outer and inner parts for clarity.

Learn how to differentiate the function y = e^(7-x) using the chain rule. This problem explores exponential functions and how to apply the chain rule for differentiation. Suitable for students in Grades 11-12 or introductory college calculus.

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