Differentiate the following w.r.t. x :
`e^(x)/x`

Answer

To differentiate the function \( y = \frac{e^x}{x} \) with respect to \( x \), we will use the quotient rule of differentiation. The quotient rule states that if you have a function in the form \( y = \frac{u}{v} \), then the derivative \( \frac{dy}{dx} \) is given by:

\[ \frac{dy}{dx} = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} \]

where \( u = e^x \) and \( v = x \).

### Step 1: Identify \( u \) and \( v \)

Let:
- \( u = e^x \)
- \( v = x \)

### Step 2: Differentiate \( u \) and \( v \)

Now we need to find \( \frac{du}{dx} \) and \( \frac{dv}{dx} \).

- The derivative of \( u \):
\[ \frac{du}{dx} = e^x \]

- The derivative of \( v \):
\[ \frac{dv}{dx} = 1 \]

### Step 3: Apply the Quotient Rule

Now we can apply the quotient rule:

\[ \frac{dy}{dx} = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} \]

Substituting the values we found:

\[ \frac{dy}{dx} = \frac{x \cdot e^x - e^x \cdot 1}{x^2} \]

### Step 4: Simplify the Expression

Now simplify the expression:

\[ \frac{dy}{dx} = \frac{x e^x - e^x}{x^2} \]

Factor out \( e^x \):

\[ \frac{dy}{dx} = \frac{e^x (x - 1)}{x^2} \]

### Final Result

Thus, the derivative of \( y = \frac{e^x}{x} \) with respect to \( x \) is:

\[ \frac{dy}{dx} = \frac{e^x (x - 1)}{x^2} \]

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