What is the second derivative of f(x)=x^2/(x^2+3) ?

1 Answer
Jan 17, 2016

[18(1-x^2)]/(x^2+3)^3

Explanation:

Using the quotient rule which states that:

If f(x) = (u(x))/(v(x)) then f'(x) = (vu' - uv')/v^2

u = x^2, u' = 2x
v=x^2+3, v'=2x

f'(x)=((x^2+3)*2x-(x^2)*2x)/(x^2+3)^2

=(6x)/(x^2+3)^2

Reset u & v.

u=6x, u'=6
v=(x^2+3)^2, v'=4x(x^2+3)

f''(x)=((x^2+3)^2*6-6x*4x(x^2+3))/(x^2+3)^4

=((x^2+3)[6x^2+18-24x^2])/(x^2+3)^4=[18(1-x^2)]/(x^2+3)^3