What is the derivative of the function f(x) = 5?
There are at least two ways to do this.
Way 1: Remember that "derivative" is the same thing as "slope." If we graph f(x) = 5, we find a horizontal line:
What is the slope of this horizontal line? Zero, of course (at every value of x). Therefore f'(x) = 0.
Way 2: Using the limit definition of the derivative,
Since f is 5 everywhere, regardless of its input,
Make it rain.