Part 2/4:
To visualize this, consider plotting the function on a graph where the ( y )-axis represents ( f(x) ) and the ( x )-axis represents ( x ). The graph of a constant function is represented as a horizontal line. For example, if ( C = 3 ), the graph would be a straight line intersecting the ( y )-axis at ( y = 3 ).
It is in this horizontal line that we find our first clue about the derivative: the slope of a horizontal line is always zero.
Slope and Derivative
The derivative of a function at a particular point is defined as the slope of the tangent line to the curve at that point. In mathematical terms, the slope is calculated using the formula:
[
\text{slope} = \frac{\text{rise}}{\text{run}}.
]