Question

Find the derivative of y=3x/(x^2 + 1)

Answer 1

To find the derivative of

`y=(3x)/(x^2+1),`

we will use the division rule for derivatives.

The division rule states that:

`((f(x))/(g(x)))^(\prime)=(f^(\prime)(x)g(x)-g^(\prime)(x)f(x))/(g(x)^2).`

Therefore, the derivative of the given quotient is:

`y^(\prime)=((3x)^(\prime)(x^2+1)-(x^2+1)^(\prime)(3x))/((x^2+1)^2).`

Simplifying the above result we get:

`\begin{gathered} y^(\prime)=(3(x^2+1)-(2x\cdot3x))/((x^2+1)^2)=(3(x^2+1)-6x^2)/((x^2+1)^2)=(3(x^2+1-2x^2))/((x^2+1)^2) \\ =(3(1-x^2))/((x^2+1)^2). \end{gathered}`

Answer:

`y^(\prime)=(3(1-x^2))/((x^2+1)^2).`