# 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?

Jul 6, 2015

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:
$f \left(x\right) = g \frac{x}{h \left(x\right)}$ where $h \left(x\right)$ 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:
$f \left(x\right) = \sin \frac{x}{3}$
use the Quotient Rule:
$f ' \left(x\right) = \frac{3 \cdot \cos \left(x\right) - 0}{9} =$
$= \cos \frac{x}{3}$

Otherwise you could see it as $f \left(x\right) = \frac{1}{3} \sin \left(x\right)$ deriving immediatelly as $f ' \left(x\right) = \frac{1}{3} \cos \left(x\right)$