# Thread: Quotient Rule Problem

1. ## Quotient Rule Problem

$y = \dfrac{x + 3}{x^{3} + x - 5}$

$\dfrac{dy}{dx} = \dfrac{[x^{3} + x - 5][\dfrac{d}{dx}(x + 3)] - [x + 3][\dfrac{d}{dx}(x^{3} + x - 5)]}{(x^{3} + x - 5)^{2}}$

$\dfrac{dy}{dx} = \dfrac{[x^{3} + x - 5][1] - [x + 3][3x + 1]}{(x^{3} + x - 5)^{2}}$

2. ## Re: Quotient Rule Problem

Originally Posted by Jason76
$y = \dfrac{x + 3}{x^{3} + x - 5}$

$\dfrac{dy}{dx} = \dfrac{[x^{3} + x - 5][\dfrac{d}{dx}(x + 3)] - [x + 3][\dfrac{d}{dx}(x^{3} + x - 5)]}{(x^{3} + x - 5)^{2}}$

$\dfrac{dy}{dx} = \dfrac{[x^{3} + x - 5][1] - [x + 3][3x + 1]}{(x^{3} + x - 5)^{2}}$
The derivative of $x^3$ is $3x^2$, not $3x$. (That may have been a typo.)

3. ## Re: Quotient Rule Problem

$y = \dfrac{x + 3}{x^{3} + x - 5}$

$\dfrac{dy}{dx} = \dfrac{[x^{3} + x - 5][\dfrac{d}{dx}(x + 3)] - [x + 3][\dfrac{d}{dx}(x^{3} + x - 5)]}{(x^{3} + x - 5)^{2}}$

$\dfrac{dy}{dx} = \dfrac{[x^{3} + x - 5][1] - [x + 3][3x^{2} + 1]}{(x^{3} + x - 5)^{2}}$

$\dfrac{dy}{dx} = \dfrac{[x^{3} + x - 5][1] - [x + 3][3x^{2} + 1]}{(x^{3} + x - 5)^{2}}$

$\dfrac{dy}{dx} = \dfrac{x^{3} + x - 5}{(x^{3} + x - 5)^{2}} - \dfrac{3x^{3} + x + 9x^{2} + 3}{(x^{3} + x - 5)^{2}}$

$\dfrac{dy}{dx} = \dfrac{x^{3} + x - 5}{(x^{3} + x - 5)(x^{3} + x - 5)} - \dfrac{3x^{3} + x + 9x^{2} + 3}{(x^{3} + x - 5)(x^{3} + x - 5)}$

$\dfrac{dy}{dx} = \dfrac{1}{(x^{3} + x - 5)} - \dfrac{3x^{3} + x + 9x^{2} + 3}{(x^{3} + x - 5)(x^{3} + x - 5)}$ Can this simplified further?

4. ## Re: Quotient Rule Problem

$\text{From your last line, } \frac{dy}{dx}=\frac{x^3+x-5-(3x^3+x+9x^2+3)}{(x^3+x-5)^2}=\frac{-2x^3-9x^2-8}{(x^3+x-5)^2}$

I believe this is the simplified form, since the numerator has no rational root. If someone disagrees, please inform the poster and/ or me.

5. ## Re: Quotient Rule Problem

Originally Posted by chen09
$\text{From your last line, } \frac{dy}{dx}=\frac{x^3+x-5-(3x^3+x+9x^2+3)}{(x^3+x-5)^2}=\frac{-2x^3-9x^2-8}{(x^3+x-5)^2}$

I believe this is the simplified form, since the numerator has no rational root. If someone disagrees, please inform the poster and/ or me.
It can be factored out with (x + 4.6821). I did not do this using the radical method, I got it from the graph.

6. ## Re: Quotient Rule Problem

ok, looks good. The computer says it's right. Thanks.