The Basic Rules
The functions \(f \left(\right. x \left.\right) = c\) and \(g \left(\right. x \left.\right) = x^{n}\) where \(n\) is a positive integer are the building blocks from which all polynomials and rational functions are constructed. To find derivatives of polynomials and rational functions efficiently without resorting to the limit definition of the derivative, we must first develop formulas for differentiating these basic functions.
The Constant Rule
We first apply the limit definition of the derivative to find the derivative of the constant function, \(f \left(\right. x \left.\right) = c .\) For this function, both \(f \left(\right. x \left.\right) = c\) and \(f \left(\right. x + h \left.\right) = c ,\) so we obtain the following result:
The rule for differentiating constant functions is called the constant rule. It states that the derivative of a constant function is zero; that is, since a constant function is a horizontal line, the slope, or the rate of change, of a constant function is \(0 .\) We restate this rule in the following theorem.
Theorem 3.2
The Constant Rule
Let \(c\) be a constant.
If \(f \left(\right. x \left.\right) = c ,\) then \(f^{'} \left(\right. x \left.\right) = 0 .\)
Alternatively, we may express this rule as
Example 3.17
Applying the Constant Rule
Find the derivative of \(f \left(\right. x \left.\right) = 8 .\)
Solution
This is just a one-step application of the rule:
Checkpoint 3.11
Find the derivative of \(g \left(\right. x \left.\right) = −3 .\)
This lesson is part of:
Derivatives