Find the Derivative. 

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Expert Answers
lemjay eNotes educator| Certified Educator

`f(x)= x^2/(x^2-3)`

To get the derivative of this, apply the quotient rule.

`d/dx(u/v) = (v*(du)/dx - u*(dv)/dx)/v^2`

In our function, we have

`u=x^2`

and

`v= x^2-3`

If we take the derivative of u and v, we will get

`(du)/dx = d/dx(x^2)`

`(du)/dx=2x`

`(dv)/dx=d/dx(x^2-3)`

`(dv)/dx=2x`

Then, plug-in them to the derivative formula above.

`f'(x)= (v*(du)/dx - u*(dv)/dx)/v^2`

`f'(x)=((x^2-3)(2x) - x^2(2x))/(x^2-3)^2`

And, simplify it.

`f'(x) = (2x^3-6x - 2x^3)/(x^2-3)^2`

`f'(x) = -(6x)/(x^2-3)^2`

Therefore, the derivative of the given function is `f'(x) =-(6x)/(x^2-3)^2` .