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Derivatives of Rational Functions

April 9th, 2009 |

Author: Merrin

A rational function is the quotient of two polynomials and is defined where

$Q(x)\ne 0$

$f(x)=\cfrac{P(x)}{Q(x)} =\cfrac{a_{0} +a_{1} x+a_{2} x^{2} +...}{b_{0} +b_{1} x+b_{2} x^{2} +...}$

Rational functions are a type of differentiable elementary function by the quotient rule. From the nature of the quotient rule, we see that the derivative of a rational function is another rational function. Successive derivatives of rational functions with non constant denominators grow in complexity because each differentiation squares the denominator.

Example 1. Find the derivative of the rational function

$f(x)=\cfrac{1+ax+x^{2} }{1+bx+x^{2} }$

Solution 1. According to the quotient rule, the derivative of f(x) is

$\cfrac{d}{dx} f(x)=\cfrac{P(x)'Q(x)-Q'(x)P(x)}{[Q(x)]^{2} } \\=\cfrac{(a+2x)(1+bx+x^{2} )-(b+2x)(1+ax+x^{2} )}{(1+bx+x^{2} )^{2} } \\=\cfrac{(a+2x-b-2x)(1+x^{2} )+(a+2x)(bx)-(b+2x)(ax)}{(1+bx+x^{2} )^{2} } \\=\cfrac{(a-b)(1+x^{2} )+2bx^{2} -2ax^{2} }{(1+bx+x^{2} )^{2} } \\=\cfrac{(a-b)(1-x^{2} )}{(1+bx+x^{2} )^{2} }$