`(d)/(dx)((logx)/(x^(2)))=`

Options

A. `x^(-2)(1+2logx)`
B. `x^(-2)(2logx-1)`
C. `x^(-3)(1-2logx)`
D. `x^(-4)(1-2logx)`

Answer

To find the derivative of the function \(\frac{\log x}{x^2}\), we will use the quotient rule of differentiation. The quotient rule states that if you have a function \(\frac{u}{v}\), then its derivative is given by:

\[ \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} \]

In our case, let:
- \(u = \log x\)
- \(v = x^2\)

Now, we will find the derivatives of \(u\) and \(v\):

1. **Find \( \frac{du}{dx} \)**:
\[ \frac{du}{dx} = \frac{d}{dx}(\log x) = \frac{1}{x} \]

2. **Find \( \frac{dv}{dx} \)**:
\[ \frac{dv}{dx} = \frac{d}{dx}(x^2) = 2x \]

Now, we can apply the quotient rule:

3. **Apply the quotient rule**:
\[ \frac{d}{dx}\left(\frac{\log x}{x^2}\right) = \frac{x^2 \cdot \frac{1}{x} - \log x \cdot 2x}{(x^2)^2} \]

4. **Simplify the numerator**:
\[ = \frac{x^2 \cdot \frac{1}{x} - 2x \log x}{x^4} \]
\[ = \frac{x - 2x \log x}{x^4} \]

5. **Factor out \(x\) from the numerator**:
\[ = \frac{x(1 - 2\log x)}{x^4} \]

6. **Simplify the expression**:
\[ = \frac{1 - 2\log x}{x^3} \]

Thus, the final answer is:
\[ \frac{1 - 2\log x}{x^3} \]