# Thread: Differentiation with quotient rule

1. ## Differentiation with quotient rule

$F(x)=\frac{x^4-5x^3+\sqrt{x}}{x^2}$

solution attempt: Book asks one to to try solving two ways once with quotient rule and once simplifying first. I have shown here only the quotient rule solution as that is giving me enough trouble to warrant a post

With quotient rule,

$[(4x^3-15x^2+\frac{1}{2}x^\expfrac{-1/2})(x^2)] - [(2x)(x^4-5x^2+\sqrt{x})]$

$= [4x^5-5x^4+\frac{1}{2} x^\expfrac{3/2}] - [2x^5-10x^4+2x(x)^\expfrac{1/2}]$

$= \frac{2x^5+5x^4+\frac{1}{2}x^\expfrac{3/2}-2x(x)^\expfrac{1/2}}{x^4}$

this is where I am reluctant to continue as I'm unsure how to go about simplifying the exponents, in doing so I end up with:

$\frac{2x+5+\frac{1}{2}x-2x}{x^6}$

I could eliminate the $2x$ but I get the distinct feeling I performed an error 2 expressions ago... please show me my error

2. Originally Posted by Foxlion
$F(x)=\frac{x^4-5x^2+\sqrt{x}}{x^2}$

$[(4x^3-15x^2+\frac{1}{2}x^\expfrac{-1/2})(x^2)] - [(2x)(x^4-5x^2+\sqrt{x})]$
That -15x^2... What's the derivatvie of -5x^2?

-Dan

3. typo in first expression, sorry

4. Originally Posted by Foxlion

$[(4x^3-15x^2+\frac{1}{2}x^\expfrac{-1/2})(x^2)] - [(2x)(x^4-5x^2+\sqrt{x})]$

$= [4x^5-5x^4+\frac{1}{2} x^\expfrac{3/2}] - [2x^5-10x^4+2x(x)^\expfrac{1/2}]$
Okay, let's look then at

$(2x)(x^4-5x^2+\sqrt{x})$

Look at the second term:
$(2x)(-5x^2)$ is equal to what?

-Dan

5. Originally Posted by Foxlion
$F(x)=\frac{x^4-5x^3+\sqrt{x}}{x^2}$

solution attempt: Book asks one to to try solving two ways once with quotient rule and once simplifying first. I have shown here only the quotient rule solution as that is giving me enough trouble to warrant a post

With quotient rule,

$[(4x^3-15x^2+\frac{1}{2}x^\expfrac{-1/2})(x^2)] - [(2x)(x^4-5x^2+\sqrt{x})]$

$= [4x^5-5x^4+\frac{1}{2} x^\expfrac{3/2}] - [2x^5-10x^4+2x(x)^\expfrac{1/2}]$

$= \frac{2x^5+5x^4+\frac{1}{2}x^\expfrac{3/2}-2x(x)^\expfrac{1/2}}{x^4}$

this is where I am reluctant to continue as I'm unsure how to go about simplifying the exponents, in doing so I end up with:

$\frac{2x+5+\frac{1}{2}x-2x}{x^6}$
Surely that's not what was intended! $\frac{x^4-5x^3+\sqrt{x}}{x^2}= \frac{x^4}{x^2}- \frac{5x^3}{x^2}+ \frac{x^{1/2}}{x^2}= x^2- 5x+ x^{-3/2}$ is much simpler.

I could eliminate the $2x$ but I get the distinct feeling I performed an error 2 expressions ago... please show me my error