Differentiate `ax^2+b`

Options

A. `-2ax`
B. 2ax
C. 2ax+b
D. ax+2b

Answer

To differentiate the function \( ax^2 + b \), we will follow these steps:

1. **Identify the function**: Let \( y = ax^2 + b \).

2. **Differentiate with respect to \( x \)**: We will apply the differentiation rules. The derivative of a constant is zero, and for the term \( ax^2 \), we will use the power rule.

3. **Apply the power rule**: The power rule states that if \( f(x) = x^n \), then \( f'(x) = nx^{n-1} \). Here, \( a \) is a constant, and we differentiate \( x^2 \):
\[ \frac{d}{dx}(ax^2) = a \cdot \frac{d}{dx}(x^2) = a \cdot 2x = 2ax \]

4. **Differentiate the constant term**: The derivative of \( b \) (a constant) is:
\[ \frac{d}{dx}(b) = 0 \]

5. **Combine the results**: Now, we combine the derivatives of both terms:
\[ \frac{dy}{dx} = 2ax + 0 = 2ax \]

Thus, the derivative of the function \( ax^2 + b \) is:
\[ \frac{dy}{dx} = 2ax \]