Differentiate the following w.r.t. as indicated :
`e^(2x)" w.r.t. "e^(x)`

Answer

To differentiate \( e^{2x} \) with respect to \( e^{x} \), we can follow these steps:

### Step 1: Define the functions
Let:
- \( u = e^{2x} \)
- \( v = e^{x} \)

### Step 2: Differentiate \( u \) with respect to \( x \)
Using the chain rule:
\[ \frac{du}{dx} = \frac{d}{dx}(e^{2x}) = e^{2x} \cdot \frac{d}{dx}(2x) = 2e^{2x} \]

### Step 3: Differentiate \( v \) with respect to \( x \)
Similarly, we differentiate \( v \):
\[ \frac{dv}{dx} = \frac{d}{dx}(e^{x}) = e^{x} \]

### Step 4: Use the formula for differentiating with respect to another variable
We need to find \( \frac{du}{dv} \):
\[ \frac{du}{dv} = \frac{du/dx}{dv/dx} = \frac{2e^{2x}}{e^{x}} \]

### Step 5: Simplify the expression
Now, simplify \( \frac{2e^{2x}}{e^{x}} \):
\[ \frac{du}{dv} = 2e^{2x - x} = 2e^{x} \]

### Final Result
Thus, the derivative of \( e^{2x} \) with respect to \( e^{x} \) is:
\[ \frac{du}{dv} = 2e^{x} \]

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