How do I find the derivative of a fraction?

1 Answer
May 1, 2018

We use quotient rule as described below to differentiate algebraic fractions or any other function written as quotient or fraction of two functions or expressions

Explanation:

When we are given a fraction say f(x)=(3-2x-x^2)/(x^2-1). This comprises of two fractions - say one g(x)=3-2x-x^2 in numerator and the other h(x)=x^2-1, in the denominator. Here we use quotient rule as described below.

Quotient rule states if f(x)=(g(x))/(h(x))

then (df)/(dx)=((dg)/(dx)xxh(x)-(dh)/(dx)xxg(x))/(h(x))^2

Here g(x)=3-2x-x^2 and hence (dg)/(dx)=-2-2x and as h(x)=x^2-1, we have (dh)/(dx)=2x and hence

(df)/(dx)=((-2-2x)xx(x^2-1)-2x xx(3-2x-x^2))/(x^2-1)^2

= (-2x^3-2x^2+2x+2-6x+4x^2+2x^3)/(x^2-1)^2

= (2x^2-4x+2)/(x^2-1)^2

or (2(x-1)^2)/(x^2-1)^2

= 2/(x+1)^2

Observe that (3-2x-x^2)/(x^2-1)=((1-x)(3+x))/((x+1)(x-1))=(-3-x)/(x+1) and using quotient rule

(df)/(dx)=(-(x+1)-(-3-x))/(x+1)^2=2/(x+1)^2