According to Boyle's law

Options

A. `(dP//dV)_(T) = K//V^(2)`
B. `(dP//dV)_(T) = - K//V`
C. `(dV//dP)_(T) = - K//P^(2)`
D. `(dV//dP)_(T) = - K//P`

Explanation

**Step-by-Step Solution:**

1. **Understanding Boyle's Law**: Boyle's Law states that at constant temperature, the pressure (P) of a gas is inversely proportional to its volume (V). Mathematically, this can be expressed as:
\[ P \propto \frac{1}{V} \]
or
\[ PV = k \]
where \( k \) is a constant.

2. **Rearranging the Equation**: From the relationship \( PV = k \), we can express volume in terms of pressure:
\[ V = \frac{k}{P} \]

3. **Differentiating with Respect to Pressure**: We need to differentiate the volume with respect to pressure. To do this, we rewrite the volume equation:
\[ V = kP^{-1} \]
Now, we differentiate both sides with respect to \( P \):
\[ \frac{dV}{dP} = k \cdot \frac{d}{dP}(P^{-1}) \]

4. **Applying the Power Rule**: Using the power rule of differentiation, where \( \frac{d}{dx}(x^n) = nx^{n-1} \):
\[ \frac{d}{dP}(P^{-1}) = -1 \cdot P^{-2} = -\frac{1}{P^2} \]
Therefore:
\[ \frac{dV}{dP} = k \left(-\frac{1}{P^2}\right) = -\frac{k}{P^2} \]

5. **Final Expression**: Since the temperature is constant, we can represent the relationship as:
\[ \frac{dV}{dP} = -\frac{k}{P^2} \]

6. **Identifying the Correct Option**: Based on the derived equation, we can check the options provided in the question to find the correct one. The correct option should match the derived expression.