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

1 Answer
Oct 11, 2017

#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.