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.