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

Oct 11, 2017

$f ' \left(x\right) = 0$

Explanation:

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

1. Basic Principles

For $f \left(x\right) = c$, where c is a constant

$f \left(x + h\right) = c$

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

$f ' \left(x\right) = {\lim}_{h \to 0} \frac{f \left(x + h\right) - f \left(x\right)}{h} = {\lim}_{h \to 0} \frac{c - c}{h}$

$= {\lim}_{h \to 0} 0 = 0$ for all values of $c$

2. Power Rule

$f \left(x\right) = c$ could also be written as:

$f \left(x\right) = c \cdot 1 = c \cdot {x}^{0}$

The Power Rule states that for all $f \left(x\right) = {x}^{n}$,

$f ' \left(x\right) = n {x}^{n - 1}$

For $n = 0$, our equation becomes:

$f ' \left(x\right) = c \cdot n {x}^{n - 1} = c \cdot \left(0\right) {x}^{- 1} = 0$ for all values of $c$

Another way to view this is by graphing the function of $f \left(x\right) = 3$:

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.