Question

Find the derivative of f(x)= (x)/(x^2-1)?

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

Answer 1

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

Explanation:

Whenever we're finding the derivative of a fraction, it helps to use the Quotient Rule. First let's define the numerator and denominator:

`f(x)=x`
`g(x)=x^2-1`

Given this, the Quotient Rule tells us that the derivative will be equal to:

`(f'(x)*g(x)-f(x)*g'(x))/g(x)^2`

Let's find the derivative of `f(x)` and `g(x)`:
`f'(x)=1`
`g'(x)=2x` (Using the Power Rule; Derivative of a constant is 0)

Now that we know the functions and their derivatives, we can plug them into our expression for the Quotient Rule. The derivative is equal to:

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

Which is equal to:

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

Which is equal to:

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

If this was still confusing, I encourage you to Google "Quotient Rule" and get some practice at it on a website such as Khan Academy, or search it on Wikipedia. Links provided below:

https://www.khanacademy.org/

https://en.wikipedia.org/wiki/Quotient_rule