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

Answer

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

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

where \( u = \log x \) and \( v = x \).

### Step 1: Identify \( u \) and \( v \)
Let:
- \( u = \log x \)
- \( v = x \)

### Step 2: Differentiate \( u \) and \( v \)
Now, we need to find \( \frac{du}{dx} \) and \( \frac{dv}{dx} \):
- \( \frac{du}{dx} = \frac{1}{x} \) (the derivative of \( \log x \))
- \( \frac{dv}{dx} = 1 \) (the derivative of \( x \))

### Step 3: Apply the Quotient Rule
Now we can apply the quotient rule:

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

### Step 4: Simplify the Expression
Now simplify the expression:

1. The first term becomes \( x \cdot \frac{1}{x} = 1 \).
2. The second term is \( \log x \cdot 1 = \log x \).

So we have:

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

### Final Result
Thus, the derivative of \( y = \frac{\log x}{x} \) with respect to \( x \) is:

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

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