Question

How do you differentiate `f(x)=e^x/(1-x^2)` using the quotient rule?

Answer 1

The quotient rule is: `((f)/(g))^'(x)=(f^'(x)*g(x)-f(x)*g^'(x))/(g^2(x))`
where `g(x)!=0`.

Explanation:

`f(x)=e^x` , `g(x)=1-x^2`
Calculate the derivatives
`f^'(x)=e^x`
`g^'(x)=(1-x^2)^'=1^'-(x^2)^'=0-2*x=-2x`
Use quotient rule:
`((f)/(g))^'(x)=(e^x*(1-x^2)-e^x*(-2x))/((1-x^2)^2)=`
`=e^x*(-x^2+2x+1)/((1-x^2)^2)`