How can you derive the quotient rule?

1 Answer
Aug 19, 2014

This can be proven fairly quickly, assuming knowledge of prior subjects such as the product rule and chain rule. Suppose #f(x) = (u(x))/(v(x))#. As we know that all of our equations are in terms of #x#, henceforth #x# will be omitted from the steps below. Note however that it is still present as the variable for the functions.

#(d/dx)f = (d/dx)u/v#

Then via our definition #f= u/v# we get #u= f*v#. Differentiating this via use of the product rule nets us...

#u' = f'*v + f*v'#

Now as we isolate f' on its own side...

#f'= [u'-f*v']/(v)#

Recalling that #f=u/v# this becomes...

#f' = [u' - (u/v)*v']/v#

And by multiplying both the numerator and denominator by #v# we get...

#f' = [u'*v - u*v']/[v^2]#

Or, by showing #x# again...

#f'(x) = [u'(x)*v(x) - u(x)*v'(x)]/(v(x))^2#