Find `(a)dy/dxand(b)(d^(2)y)/(dx^(2))` when y is given by :
`logx-x`

Answer

To solve the problem, we need to find the first and second derivatives of the function \( y = \log x - x \).

### Step 1: Find the first derivative \( \frac{dy}{dx} \)

Given:
\[ y = \log x - x \]

To find \( \frac{dy}{dx} \), we differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = \frac{d}{dx}(\log x) - \frac{d}{dx}(x) \]

Using the differentiation rules:
- The derivative of \( \log x \) is \( \frac{1}{x} \).
- The derivative of \( x \) is \( 1 \).

Thus:
\[ \frac{dy}{dx} = \frac{1}{x} - 1 \]

### Step 2: Find the second derivative \( \frac{d^2y}{dx^2} \)

Now, we need to differentiate \( \frac{dy}{dx} \) to find \( \frac{d^2y}{dx^2} \):
\[ \frac{d^2y}{dx^2} = \frac{d}{dx}\left(\frac{1}{x} - 1\right) \]

Differentiating each term:
- The derivative of \( \frac{1}{x} \) is \( -\frac{1}{x^2} \).
- The derivative of \( -1 \) is \( 0 \).

Thus:
\[ \frac{d^2y}{dx^2} = -\frac{1}{x^2} \]

### Final Answers:
(a) \( \frac{dy}{dx} = \frac{1}{x} - 1 \)

(b) \( \frac{d^2y}{dx^2} = -\frac{1}{x^2} \)

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