**Problem : **
Find the derivative of the function *f* (*x*) = 5*x* + 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*)