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:

Figure %: Plot of f (x)

Figure %: Derivative of f (x), f'(x)