Find `dy/dx` for`(ax^(2)+ b/x)`

Answer

To find \( \frac{dy}{dx} \) for the function \( y = ax^2 + \frac{b}{x} \), we will differentiate the function with respect to \( x \).

### Step-by-Step Solution:

1. **Identify the Function**:
\[ y = ax^2 + \frac{b}{x} \]

2. **Rewrite the Function**:
We can rewrite \( \frac{b}{x} \) as \( b \cdot x^{-1} \):
\[ y = ax^2 + b \cdot x^{-1} \]

3. **Differentiate Each Term**:
We will differentiate \( y \) with respect to \( x \) using the power rule \( \frac{d}{dx}(x^n) = n \cdot x^{n-1} \):
- The derivative of \( ax^2 \) is:
\[ \frac{d}{dx}(ax^2) = a \cdot 2x = 2ax \]
- The derivative of \( b \cdot x^{-1} \) is:
\[ \frac{d}{dx}(b \cdot x^{-1}) = b \cdot (-1) \cdot x^{-2} = -\frac{b}{x^2} \]

4. **Combine the Derivatives**:
Now, we combine the results from the differentiation:
\[ \frac{dy}{dx} = 2ax - \frac{b}{x^2} \]

5. **Final Result**:
Thus, the derivative \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = 2ax - \frac{b}{x^2} \]