Question

How do you find the derivative of `f(x) = 3`?

Answer 1

`f'(x) = 0`

Explanation:

The derivative of a constant is always `0`. To prove this, there are different methods you can take.

1. Basic Principles

For `f(x) = c`, where c is a constant

`f(x + h) = c`

(Normally you would replace every instance of `x` with `(x+h)`, but since there is only a constant and no `x`, it remains just `c`).

`f'(x) = lim_(h->0) (f(x+h) - f(x))/h = lim_(h->0)(c-c)/h`

`= lim_(h->0) 0 = 0` for all values of `c`

2. Power Rule

`f(x) = c` could also be written as:

`f(x) = c*1 = c*x^0`

The Power Rule states that for all `f(x) = x^n`,

`f'(x) = nx^(n-1)`

For `n = 0`, our equation becomes:

`f'(x) = c*nx^(n-1) = c*(0)x^(-1) = 0` for all values of `c`

Another way to view this is by graphing the function of `f(x) = 3`:

graph{3*x^0 [-10, 10, -5, 15]}

The derivative of a function can also be viewed as the slope of that function at a certain point in time. Since a constant value is graphed as a straight line and never moves up or down, the slope will always be `0`. This will still be true for any value of `c`, since they will still be a straight line when graphed, though intersecting at a different point on the y-axis.