Calculus: how to apply Derivative Rules?

[oleksiy]

Hi everybody, I am newcomer here .

Trying to solve a problem, got lost with multiple rules of derivatives.



I solved the numerator, using the Product Rule.

Got: 8x³ + 24x² - 24x - 48. This is correct.

Which rule must I use next? My guess was the Quotient Rule:



Using obtained numerator as f(x) and the initial denominator as g(x). But it looks like it is wrong approach.

Please advise! Many thanks!

Oleksiy

 

[LCKurtz]

There is nothing wrong with that approach. Alternatively you could have multiplied the numerator out first and then differentiated it to get the same polynomial in the numerator. Then an approach you could use is
[MATH]\frac{8x^3+24x^2-24x-48}{x^5} = 8x^{-2}+24x^{-3}-24x^{-4}-48x^{-5}[/MATH]and differentiate it as a polynomial. But then you would likely want to simplify with common denominator, which isn't too bad.
[Edit] Please ignore this post. It didn't say what I meant to say, see the next post.

 

[Dr.Peterson]

I solved the numerator, using the Product Rule.

Got: 8x³ + 24x² - 24x - 48. This is correct.

Which rule must I use next? My guess was the Quotient Rule:

View attachment 12697

Using obtained numerator as f(x) and the initial denominator as g(x). But it looks like it is wrong approach.

I'm not sure whether what you described is the correct work or not; it depends on what you actually did. You have shown too few steps.

But first, what is the goal? Am I right that you want to differentiate f? You didn't actually say that; the word "solve" is inappropriate.

If you used the product rule, I would expect you to have obtained a different form than you show; you could just as well have expanded the numerator ("multiplied it out") and differentiated in that form, which is easier in this case. And if you do that, you might as well carry out the division by x^5 (before differentiating anything) and differentiate the polynomial that results.

At this point you have differentiated the numerator only; if you choose to use the quotient rule, you have to put that into the rule (where you are calling this f'(x)) along with f(x), g(x), and g'(x). Please show your work at that point, so we can check out the details.

What LCKurtz did appears to be dividing the derivative of the numerator by the denominator, rather than dividing the expanded numerator itself. I think that was a mistake.

 

[LCKurtz]

What LCKurtz did appears to be dividing the derivative of the numerator by the denominator, rather than dividing the expanded numerator itself. I think that was a mistake.

You're right. I misunderstood what he said.

 

[JeffM]

You can use the product rule in conjunction with the quotient rule. It will give you the correct answer, but it is messy indeed.

[MATH]f(x) = \dfrac{p(x) * q(x)}{v(x)} \implies f'(x) = \dfrac{\{p(x) * q'(x) + p'(x) * q(x)\} * v(x) - p(x) * q(x) * v'(x)}{\{v(x)\}^2}.[/MATH]
It is, however, far less error prone to expand

[MATH]p(x) * q(x) = u(x)[/MATH] as step 1. Then simply apply the quotient rule.

[MATH]\therefore f(x) = \dfrac{u(x)}{v(x)} \implies f'(x) = \dfrac{u'(x) * v(x) - u(x) * v'(x)}{\{v(x)\}^2}.[/MATH]
Note that to use the first method requires you to calculate p(x) * q(x) in any case.

 

[pka]

Hi everybody, I am newcomer here .

Trying to solve a problem, got lost with multiple rules of derivatives.

View attachment 12696

Have a look/see here.