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?

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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 \( f(x) \).

Learn how to differentiate e^{2x-6} using the chain rule. The solution involves understanding the chain rule and applying it step-by-step to find the derivative. Suitable for students in Grades 11-12 studying calculus.

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