Question

What is the derivative of f(x)= x^3-1/x-1??

Answer 1

`color(purple)((2x^3-3x^2+1)/(x-1)^2)`

Explanation:

We're dealing with a rational function here, which tells us to use the quotient rule.

Our function is made up of two functions:

`color(blue)(h(x)=x^3-1=>h'(x)=3x^2)`
`color(red)(g(x)=x-1=>g'(x)=1)`

The Quotient Rule tells us that the derivative will be:

`(h'(x)g(x)-h(x)g'(x))/g(x)^2`

Now, we just plug in. We get:

`(color(blue)(3x^2)color(red)((x-1))-color(blue)((x^3-1))color(red)(1))/color(red)((x-1)^2)`

`=>(3x^3-3x^2-x^3+1)/(x-1)^2`

`=>color(purple)((2x^3-3x^2+1)/(x-1)^2)`

Our final answer being in purple.

We just had to realize we're dealing with a rational function, so we had to use the Quotient Rule. We find our two functions, their derivative, and plug into the formula.

Hope this helps!