Find `(dy)/(dx)` for the following :
`y=x^(-3)`

Answer

To find the derivative \(\frac{dy}{dx}\) for the function \(y = x^{-3}\), we will use the power rule of differentiation. The power rule states that if \(y = x^n\), then the derivative \(\frac{dy}{dx} = n \cdot x^{n-1}\).

### Step-by-Step Solution:

1. **Identify the function**:
We have \(y = x^{-3}\).

2. **Apply the power rule**:
According to the power rule, if \(y = x^n\), then:
\[ \frac{dy}{dx} = n \cdot x^{n-1} \]
Here, \(n = -3\).

3. **Differentiate**:
Substitute \(n = -3\) into the power rule:
\[ \frac{dy}{dx} = -3 \cdot x^{-3-1} \]

4. **Simplify the expression**:
Now simplify the exponent:
\[ \frac{dy}{dx} = -3 \cdot x^{-4} \]

5. **Final result**:
Thus, the derivative is:
\[ \frac{dy}{dx} = -\frac{3}{x^4} \]

### Summary of the Steps:
- Identify the function.
- Apply the power rule.
- Differentiate using the power rule.
- Simplify the expression to get the final result.