Final answer:
The derivative of a constant function f(x)=c is zero. This is because the slope of the tangent line to the graph of a constant function, which represents the function's rate of change, is always zero—indicative of no change in the value of the function as the input changes.
Step-by-step explanation:
The question asks what the derivative of a constant function f(x)=c is. According to Rule 1 of the basic rules of differentiation, the derivative of a constant function is always equal to zero. This is because the slope of the graph of a constant function is horizontal, so there is no change in the y-value as the x-value changes. Graphically, this is shown as a flat line, with no incline or decline, indicating that there is no variation in the function's output regardless of the input, hence a zero slope.
The concept of the derivative being the slope of the tangent line at a point is fundamental here. Since a constant function does not change, the slope of the tangent line is constantly zero. This follows from the definition of slope of a curve, which is the ratio of the change in the function's output (y-value) to the change in the input (x-value), symbolically given as m = Δy/Δx. With a constant output, Δy is zero, leading to a zero slope, regardless of Δx.