Differentiate `e^x` with respect to `sqrt(x))`

Answer

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

### Step 1: Define the variables
Let:
- \( y = e^x \)
- \( t = \sqrt{x} \)

We want to find \( \frac{dy}{dt} \).

### Step 2: Use the chain rule
Using the chain rule, we can express \( \frac{dy}{dt} \) as:
\[ \frac{dy}{dt} = \frac{dy}{dx} \cdot \frac{dx}{dt} \]

### Step 3: Find \( \frac{dy}{dx} \)
We know that:
\[ y = e^x \]
Differentiating \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = e^x \]

### Step 4: Find \( \frac{dt}{dx} \)
Since \( t = \sqrt{x} \), we differentiate \( t \) with respect to \( x \):
\[ \frac{dt}{dx} = \frac{1}{2\sqrt{x}} \]

### Step 5: Find \( \frac{dx}{dt} \)
To find \( \frac{dx}{dt} \), we take the reciprocal of \( \frac{dt}{dx} \):
\[ \frac{dx}{dt} = 2\sqrt{x} \]

### Step 6: Substitute into the chain rule
Now substitute \( \frac{dy}{dx} \) and \( \frac{dx}{dt} \) back into the equation from Step 2:
\[ \frac{dy}{dt} = \frac{dy}{dx} \cdot \frac{dx}{dt} = e^x \cdot (2\sqrt{x}) \]

### Step 7: Final result
Thus, the differentiation of \( e^x \) with respect to \( \sqrt{x} \) is:
\[ \frac{dy}{dt} = 2\sqrt{x} \cdot e^x \]

### Summary
The derivative of \( e^x \) with respect to \( \sqrt{x} \) is \( 2\sqrt{x} e^x \).

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