Find the derivative of f(x)`=2x^(2)+3x-4` at x=0.

Options

A. 1
B. 2
C. 3
D. 4

Explanation

To find the derivative of the function \( f(x) = 2x^2 + 3x - 4 \) at \( x = 0 \), we can use the definition of the derivative. The derivative \( f'(a) \) at a point \( a \) is given by:

\[ f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a} \]

In this case, we want to find \( f'(0) \):

1. **Calculate \( f(0) \)**:
\[ f(0) = 2(0)^2 + 3(0) - 4 = -4 \]

2. **Set up the limit for the derivative**:
\[ f'(0) = \lim_{x \to 0} \frac{f(x) - f(0)}{x - 0} = \lim_{x \to 0} \frac{f(x) + 4}{x} \]

3. **Substitute \( f(x) \)**:
\[ f(x) = 2x^2 + 3x - 4 \implies f(x) + 4 = 2x^2 + 3x - 4 + 4 = 2x^2 + 3x \]

4. **Rewrite the limit**:
\[ f'(0) = \lim_{x \to 0} \frac{2x^2 + 3x}{x} \]

5. **Simplify the expression**:
\[ f'(0) = \lim_{x \to 0} \frac{x(2x + 3)}{x} \]
Here, we can cancel \( x \) (for \( x \neq 0 \)):
\[ f'(0) = \lim_{x \to 0} (2x + 3) \]

6. **Evaluate the limit**:
\[ f'(0) = 2(0) + 3 = 3 \]

Thus, the derivative of \( f(x) \) at \( x = 0 \) is:

\[ \boxed{3} \]