Question

When you use the quotient rule to find the derivative of a function, does the denominator of the function have to have a variable or can it be a constant number?

Answer 1

Actually it is indifferent; at the end of the day you get the right answer even if you have a constant in the denominator and you use the Quotient Rule.

Explanation:

The denominator may or may not contain the variable you are deriving for.
Consider that the Quotient Rule deals specifically with this case, i.e., the derivation of a function such as:
`f(x)=g(x)/(h(x))` where `h(x)` is a function of `x`, but a constant, such as `3`, is actually a function of `x` as well (`y=3`)!

In reality you could use the Quotient Rule even if in the denominator you have a constant (it takes only a bit more...steps):
for example:
`f(x)=sin(x)/3`
use the Quotient Rule:
`f'(x)=(3*cos(x)-0)/9=`
`=cos(x)/3`

Otherwise you could see it as `f(x)=1/3sin(x)` deriving immediatelly as `f'(x)=1/3cos(x)`