If `y=ax^2+b`, then dy/dx at x=2 is equal to:

Options

A. 4a
B. 3a
C. 2a
D. None of these

Answer

To solve the problem, we need to find the derivative of the function \( y = ax^2 + b \) with respect to \( x \) and then evaluate it at \( x = 2 \).

### Step-by-Step Solution:

1. **Identify the function**: We have the function given as:
\[ y = ax^2 + b \]

2. **Differentiate the function**: We need to find the derivative \( \frac{dy}{dx} \). The derivative of \( ax^2 \) with respect to \( x \) can be found using the power rule of differentiation, which states that if \( y = x^n \), then \( \frac{dy}{dx} = nx^{n-1} \).
- For \( ax^2 \), the derivative is:
\[ \frac{d}{dx}(ax^2) = 2ax^{2-1} = 2ax \]
- The derivative of \( b \) (a constant) is:
\[ \frac{d}{dx}(b) = 0 \]
- Therefore, combining these results, we have:
\[ \frac{dy}{dx} = 2ax + 0 = 2ax \]

3. **Evaluate the derivative at \( x = 2 \)**: Now we need to substitute \( x = 2 \) into the derivative:
\[ \frac{dy}{dx} \bigg|_{x=2} = 2a(2) = 4a \]

4. **Conclusion**: Thus, the value of \( \frac{dy}{dx} \) at \( x = 2 \) is:
\[ \frac{dy}{dx} \bigg|_{x=2} = 4a \]

### Final Answer:
\[ \frac{dy}{dx} \text{ at } x = 2 \text{ is } 4a \]

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