What is the derivative of `e^(x)` with respect to `x^(e )` ?

Options

A. `(xe^(x))/(ex^(e ))`
B. `(e^(x))/(x^(e ))`
C. `(xe^(x))/(x^(e ))`
D. `(e^(x))/(ex^(e ))`

Answer

To find the derivative of \( e^x \) with respect to \( x^e \), we can use the concept of implicit differentiation. Let's go through the steps:

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

### Step 2: Differentiate \( u \) and \( v \)
Now we need to find the derivatives \( \frac{du}{dx} \) and \( \frac{dv}{dx} \).

1. Differentiate \( u \):
\[ \frac{du}{dx} = \frac{d}{dx}(e^x) = e^x \]

2. Differentiate \( v \):
Using the power rule, we have:
\[ \frac{dv}{dx} = \frac{d}{dx}(x^e) = e \cdot x^{e-1} \]

### Step 3: Apply the formula for the derivative of \( u \) with respect to \( v \)
We use the formula:
\[ \frac{du}{dv} = \frac{du/dx}{dv/dx} \]
Substituting the derivatives we found:
\[ \frac{du}{dv} = \frac{e^x}{e \cdot x^{e-1}} \]

### Step 4: Simplify the expression
Now, we can simplify the expression:
\[ \frac{du}{dv} = \frac{e^x}{e \cdot x^{e-1}} = \frac{e^{x-1}}{x^{e-1}} \]

### Final Result
Thus, the derivative of \( e^x \) with respect to \( x^e \) is:
\[ \frac{du}{dv} = \frac{e^x}{e \cdot x^{e-1}} \]