The derivative of a quotient is not the derivative of the
numerator divided by the derivative of the denominator. The video
below shows this with an example. Instead, we have
The Quotient
Ruleddx(f(x)g(x))=g(x)f′(x)−f(x)g′(x)(g(x))2.
The derivative of the quotient is not
the quotient of the derivatives.
We write, briefly, (fg)′=gf′−fg′g2=lo
de hi−hi de lololo, where
hi=numerator, lo=denominator, and de=differentiate.
The important thing to remember here is that unlike the product
rule, where f′g+fg′=fg′+f′g and the order doesn't matter,
gf′−fg′≠fg′−gf′. Lo comes first!
The quotient rule can be derived from the product rule. If we
write f(x)=g(x)f(x)g(x), then the
product rule says that f′(x)=(g(x)⋅f(x)g(x))′; i.e, f′(x)=g′(x)f(x)g(x)+g(x)(f(x)g(x))′.
Solving for (f(x)g(x))′ gives (f(x)g(x))′=f′(x)−g′(x)f(x)g(x)g(x)=g(x)f′(x)−f(x)g′(x)[g(x)]2.