`e^(x/a)`

Answer

To find the differential coefficient of the function \( e^{(x/a)} \) with respect to \( x \), we will use the chain rule of differentiation. Here are the steps to solve the problem:

### Step 1: Identify the function
The function we need to differentiate is:
\[ y = e^{(x/a)} \]

### Step 2: Apply the chain rule
The chain rule states that if you have a composite function \( y = e^{u} \) where \( u = \frac{x}{a} \), then the derivative \( \frac{dy}{dx} \) can be found using:
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \]

### Step 3: Differentiate the outer function
The derivative of \( e^{u} \) with respect to \( u \) is:
\[ \frac{dy}{du} = e^{u} \]

### Step 4: Differentiate the inner function
Next, we differentiate \( u = \frac{x}{a} \) with respect to \( x \):
\[ \frac{du}{dx} = \frac{1}{a} \]

### Step 5: Combine the derivatives
Now, we can combine the results from Step 3 and Step 4:
\[ \frac{dy}{dx} = e^{u} \cdot \frac{du}{dx} = e^{(x/a)} \cdot \frac{1}{a} \]

### Step 6: Write the final answer
Thus, the differential coefficient of \( e^{(x/a)} \) with respect to \( x \) is:
\[ \frac{dy}{dx} = \frac{1}{a} e^{(x/a)} \]