Subject: Calculus

Constant Factor Rule In Differentiation


What are constant functions?

Some of the functions in Calculus involves constants. These are functions that does not depend on the independent variable. For example, a function f(x) has a dependent variable x. So when we say "does not depend", it means any number, letters or symbols that does not include x is considered a constant function. Thus, functions f(x)=100 and f(x)=k are classified as constant functions while f(x)=x^{2} is not.

Okay I get it, then what is its derivative?

We can easily present here what is the derivative of a constant function but before telling you that, let us first answer the questions "WHY" and "HOW". We know in the previous sections that derivative is based on the processes of the concepts of limits. Particularly, we solved derivative of a function using the "three-step rule". We'll try to get now the derivative of a constant function using the three-step rule.
Suppose that f(x)=s, then the derivative of f is given by

First Step

f(x+h)-f(x)=s-s=0.

Second Step

\frac{0}{h}=0.

Third Step

\lim_{h \to 0}0=0.

Therefore, according to the three-step rule, the derivative of a constant is zero.

Example #1

Find \frac{d}{dz}f when f is f(x)=4x.

Note that our variable of differentiation is z, thus the 4x is a constant. whose derivative is 0.

Example #2

Find the first derivative of f when f(c)=c.

Most of the students treat the letter c as constants. But for the function f(c), c is not anymore a constant. It is now our variable of differentiation. Therefore, its derivative is not zero but 1. 1

Example #3

Find f'(y) when f(y)=2x^2.

For the function defined by f(y)=2x^2, it is a constant function, thus, its derivative is

0

Example #4

Find the derivative of f when f(x)=0.

0 is just a constant whose derivative is zero. Therefore, the derivative of f is 0.

Example #5

Find the derivative of f when f(x)=\cos{t}.

t is just a constant and so is \cos{t} Therefore, the derivative of f is 0.

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