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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,

Make it rain.