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?

1 Answer
Gió
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(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)#

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