The given function is $f : x \mapsto e^{2x - 7}$, and we are asked to find its derivative.

To differentiate this function, we apply the chain rule, which states that the derivative of $e^{g(x)}$ is $e^{g(x)} \cdot g'(x)$, where $g(x)$ is the inner function.

In our case, $g(x) = 2x - 7$, so let's compute the derivative:

1. The derivative of $g(x) = 2x - 7$ is $g'(x) = 2$.

Now, applying the chain rule: $f'(x) = e^{2x - 7} \cdot 2 = 2e^{2x - 7}.$

Thus, the derivative of the function is: $f'(x) = 2e^{2x - 7}.$

Would you like a more detailed explanation or have any further questions?

Here are 5 related questions you might find helpful:

1. How do you apply the chain rule to more complex functions?
2. What is the derivative of other exponential functions like $e^{x^2}$?
3. How can the chain rule be used in multivariable calculus?
4. What is the geometric interpretation of the derivative of an exponential function?
5. How does the constant $e$ influence the behavior of the function and its derivative?

Tip: Always remember to first identify the inner function when applying the chain rule.