# Finding a f'(x) in a fraction

• May 29th 2009, 04:19 PM
hemi
Finding a f'(x) in a fraction
I don't know how I should go about solving down on this problem for $\displaystyle f'(x)$:

$\displaystyle f(x)=\frac{-6x^3+3x^2-2x+1}{x}$

If someone could show me in a step-by-step fashion that would be great. :)
• May 29th 2009, 04:23 PM
Random Variable
You could use the quotient rule or you could just divide each term in the numerator by x and then take the derivative.
• May 29th 2009, 04:23 PM
Spec
Quotient rule - Wikipedia, the free encyclopedia

In this case, it's easier to just split it up.

$\displaystyle f(x)=\frac{-6x^3+3x^2-2x+1}{x}=-6x^2+3x-2+\frac{1}{x}$
• May 29th 2009, 07:45 PM
hemi
I'm still a bit confused on the quotient rule, can you go a bit further with the math explaination? :)

Thanks a lot!
• May 29th 2009, 08:12 PM
Random Variable
Quote:

Originally Posted by hemi
I'm still a bit confused on the quotient rule, can you go a bit further with the math explaination? :)

Thanks a lot!

For this problem there is no reason to use the quotient rule. But here's what it would look like:

if $\displaystyle f(x) = \frac {g(x)}{h(x)}$ then $\displaystyle f'(x) = \frac {h(x)g'(x)-g(x)h'(x)}{[h(x)]^{2}}$

so $\displaystyle f'(x) = \frac {x \frac {d}{dx} (-6x^{3}+3x^{2}-2x+1) - (-6x^3+3x^{2}-2x+1) \frac {d}{dx}x} {x^{2}}$

$\displaystyle = \frac {x(-18x^{2}+6x-2) - (-6x^{3}+3x^{2}-2x+1)}{x^{2}}$

$\displaystyle = \frac {-12x^{3} +3x^{2}-1}{x^{2}}$

$\displaystyle = -12x + 3 - \frac {1}{x^{2}}$

The quotient rule is just an extension of the product rule.