Final answer:
The derivative of a constant is always 0, Therefore, the correct answer is: B) Derivative: 0
Step-by-step explanation:
Derivative of a Constant:
To find the derivative of a constant function, such as f(x) = 4, using the limit definition of a derivative, we apply the following formula: If f(x) is a function, then its derivative, f'(x), can be defined as: \[ f'(x) = \lim_{{h \to 0}} \frac{{f(x + h) - f(x)}}{h} \]
Now, since f(x) = 4 is a constant function, f(x + h) will also be equal to 4 for any value of h. So let's plug this into our formula: \[ f'(x) = \lim_{{h \to 0}} \frac{{4 - 4}}{h} \] \[ f'(x) = \lim_{{h \to 0}} \frac{0}{h} \] \[ f'(x) = \lim_{{h \to 0}} 0 \] As the limit approaches 0, the expression above is simply 0 because the numerator of the fraction is always 0 regardless of the value of h (except for h = 0
where the original expression is undefined, but that is not an issue when taking limits). Thus, the derivative of the constant function f(x) = 4 is: \[ f'(x) = 0 \] Therefore, the correct answer is: B) Derivative: 0