What is the derivative of the function *f*(*x*) = 5?

Answer

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,