The quotient Rule for Derivatives

quotient rule, rational functions, quotient, square

Like the product rule, the quotient rule is a way to find the derivative of a function that is the quotient of two functions.  In a way, we can think of it as the product rule applied to two functions where one of them is one over the second function.  But the more typical and readily accessible method is the quotient rule when we have a rational function, which is the same thing we mean when we say a function that is a quotient of two functions.  So, if H(x)=f(x)/g(x), then H'(x) is defined in the following way:

          f'(x)g(x)-f(x)g'(x)
H'(x)=------------------------
                g(x)²

That is, the derivative of H is the difference of the product of the derivative of the first function times the second and the product of the first function times the derivative of the second, all divided by the square of the second function.

Here's an example:

If H(x)=x² / sin(x) then, we get the following:

           2xsin(x) – x²cos(x)
H'(x)=---------------------------
                  sin(x)²

It is very useful to remember both the product and quotient rules for derivatives.

quotient rule, rational functions, quotient, square