Problem : Find the derivative of the function f (x) = 5x + 2000 .Since this is a line with slope 5 , the derivative of f is equal to 5 at every point, i.e. f'(x) = 5 . Note that the constant term does not affect the value of the derivative. This is the case for any function.
Problem : Find all functions f such that f'(x) = 0 .The condition f'(x) = 0 implies that the function must have graph with horizontal tangent at every point. The only possibility is for the graph to be a horizontal line itself. Thus the function must be f (x) = c for some constant c .
Problem : Sketch the graph of the derivative of the function f (x) with the following graph: