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#