How I do I prove the Quotient Rule for derivatives?
1 Answer
Apr 30, 2016
We can use the product rule:
#d/dx[f(x)g(x)]=f'(x)g(x)+f(x)g'(x)#
In order to prove the quotient rule, which states that
#d/dx[f(x)/g(x)]=(f'(x)g(x)-f(x)g'(x))/(g(x))^2#
However, we can apply this to the product rule by writing
Use the product rule on this:
#d/dx[f(x)(g(x))^-1]=f'(x)(g(x))^-1+f(x)d/dx[(g(x))^-1]#
Differentiating
#d/dx[(g(x))^-1]=-g'(x)(g(x))^-2#
Hence we obtain
#d/dx[f(x)(g(x))^-1]=f'(x)(g(x))^-1-f(x)g'(x)(g(x))^-2#
Rewriting with fractions, this becomes
#d/dx[f(x)/g(x)]=(f'(x))/g(x)-(f(x)g'(x))/(g(x))^2#
Rewriting with a common denominator, this becomes
#d/dx[f(x)/g(x)]=(f'(x)g(x)-f(x)g'(x))/(g(x))^2#
