Question

How do you differentiate `[(x^3-3x^2+4)/(x^2)]`?

Answer 1

`1-8/x^3`

Explanation:

Simplify the expression as `x-3+4/x^2`

Now differentiate term by term to get 1-`8/x^3`

Answer 2

You could also differentiate this expression by using the quotient rule.

Explanation:

Alternatively, you can use the quotient rule to differentiate this function.

The quotient rule tells you that a function that can be written as the quotient of two other functions

`y = f(x)/g(x)`,

can be differentiated by using this formula

`color(blue)(d/dx(y) = (f^'(x) * g(x) - f(x) * g^'(x))/[g(x)]^2)`, where `g(x) !=0`.

In your case, you have

`f(x) = x^3 - 3x^2 + 4`

and

`g(x) = x^2`

The derivative of `y` will thus be equal to

`y^' = (d/dx(x^3 - 3x^2 + 4) * x^2 - (x^3 - 3x^2 + 4) * d/dx(x^2))/(x^2)^2`

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

`y^' = (3x^4 - 6x^3 -2x^4 + 6x^3 - 8x)/x^4`

This is equivalent to

`y^' = (x^4 - color(red)(cancel(color(black)(6x^3))) + color(red)(cancel(color(black)(6x^3))) - 8x)/x^4`

`y^' = (x^4 - 8x)/x^4`

FInally, you can simplify this expression to get

`y^' = color(red)(cancel(color(black)(x^4)))/color(red)(cancel(color(black)(x^4))) - (8color(green)(cancel(color(black)(x))))/(x^(color(green)(cancel(color(black)(4))))) = 1 - 8/x^3`

If you want, you can go on to factor this by using the difference of two cubes

`color(blue)(a^3 - b^3 = (a-b) * (a^2 + ab + b^2)`

`y^' = (1)^3 - (2/x)^3 = color(green)((1-2/x) * (1 + 4/x + 4/x^2))`