To find the derivative of $e^{2x - 6}$, we can use the chain rule. The chain rule states that if $f(x) = e^{g(x)}$, then $f'(x) = e^{g(x)} \cdot g'(x)$.

Step-by-Step Solution:

1. Identify the outer function and inner function:

   • The outer function is $e^u$, where $u = 2x - 6$.
   • The inner function is $u = 2x - 6$.

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 $u = 2x - 6$ with respect to $x$ is $2$.

4. Apply the chain rule:

   • Multiply the derivative of the outer function by the derivative of the inner function: $\frac{d}{dx} \left( e^{2x - 6} \right) = e^{2x - 6} \cdot 2 = 2e^{2x - 6}$

Final Answer:

$\frac{d}{dx} \left( e^{2x - 6} \right) = 2e^{2x - 6}$

Would you like further details on this solution?

---

Here are some related questions:

1. How would you find the derivative of $e^{3x + 2}$?
2. What is the derivative of $e^{x^2 - 5x}$?
3. How do you find the second derivative of $e^{2x - 6}$?
4. What is the derivative of $e^{kx + c}$, where $k$ and $c$ are constants?
5. How does the chain rule apply to finding the derivative of $e^{f(x)}$?

Tip: Remember, the exponential function $e^{f(x)}$ differentiates by retaining itself times the derivative of $f(x)$.