Differentiate `e^(x)` w.r.t. `sqrt(x)`.

Options

A. `2e^(x) sqrt(x)`
B. `-e^(x) sqrt(x)`
C. `e^(x) sqrt(x)`
D. `2^(x) sqrt(x)`

Answer

To differentiate \( e^x \) with respect to \( \sqrt{x} \), we can use the chain rule. Here’s the step-by-step solution:

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

We want to find \( \frac{du}{dv} \).

### Step 2: Use the chain rule
According to the chain rule, we can express \( \frac{du}{dv} \) as:
\[ \frac{du}{dv} = \frac{du/dx}{dv/dx} \]

### Step 3: Differentiate \( u \) with respect to \( x \)
Now, we differentiate \( u \):
\[ \frac{du}{dx} = e^x \]

### Step 4: Differentiate \( v \) with respect to \( x \)
Next, we differentiate \( v \):
\[ v = \sqrt{x} = x^{1/2} \]
Using the power rule:
\[ \frac{dv}{dx} = \frac{1}{2} x^{-1/2} = \frac{1}{2\sqrt{x}} \]

### Step 5: Substitute into the chain rule formula
Now we substitute \( \frac{du}{dx} \) and \( \frac{dv}{dx} \) into the chain rule formula:
\[ \frac{du}{dv} = \frac{e^x}{\frac{1}{2\sqrt{x}}} \]

### Step 6: Simplify the expression
To simplify, we multiply by the reciprocal:
\[ \frac{du}{dv} = e^x \cdot 2\sqrt{x} = 2e^x\sqrt{x} \]

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

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