find the derivative f(x)=e2x+e−xf(x)=e2x+e−x

To find the derivative of the function \( f(x) = e^{2x} + e^{-x} \), we can differentiate each term separately using the rules of differentiation.

### Step-by-Step Process:

1. **Differentiate the first term: \( e^{2x} \)**:
   - The derivative of \( e^{2x} \) is \( 2e^{2x} \), because of the chain rule (the derivative of \( 2x \) is \( 2 \)).

2. **Differentiate the second term: \( e^{-x} \)**:
   - The derivative of \( e^{-x} \) is \( -e^{-x} \), again using the chain rule (the derivative of \( -x \) is \( -1 \)).

### Putting it together:
\[
f'(x) = 2e^{2x} - e^{-x}
\]

So, the derivative of the function is:
\[
f'(x) = 2e^{2x} - e^{-x}
\]

Would you like further details on the chain rule or any other part of this solution?

### Related Questions:
1. What is the second derivative of \( f(x) = e^{2x} + e^{-x} \)?
2. How do we apply the chain rule in finding derivatives?
3. What is the general form of the derivative of \( e^{ax} \)?
4. Can you integrate \( 2e^{2x} - e^{-x} \) back to find the original function?
5. How does the behavior of \( f(x) = e^{2x} + e^{-x} \) change as \( x \to \infty \)?

### Tip:
When differentiating exponential functions, always pay attention to the inner functions, as the chain rule often applies.

find the derivative f(x) = e^(2x) + e^(-x)

This detailed solution explains how to differentiate f(x) = e^(2x) + e^(-x) using the chain rule and exponential differentiation. Learn key calculus concepts and formulas, suitable for high school or introductory college calculus.

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