# How can you derive the quotient rule?

##### 1 Answer

This can be proven fairly quickly, assuming knowledge of prior subjects such as the **product rule** and **chain rule**. Suppose

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

Then via our definition

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

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

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

Recalling that

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

And by multiplying both the numerator and denominator by

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

Or, by showing

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