Differentiation Help: 2 Questions

[Hollysmoke]

There are two differentiation questions that I am working on and was wondering if I did it right and if I did it wrong, to get some help:

1) y=(2x^2-x)(3x^2+5)^8

let u = (2x^2-x) and let v=(3x^2+5)^8

y'=uv' + u'v

u'=4x-1
v'=48x(3x^2+5)^7

y'=(2x^2-x)(48x)(3x^2+5)^7+(4x-1)(3x^2+5)^8


2) This one was the one I was stuck on. I tried using quotient and product rule:

y=(t^2-6t/t^2+6t)^5

y'=5(t^2-6t/t^2+6t)^4 . 1[(2t-6)(t^2+6t)-(t^2-6)(2t+6)/(t^2+6t)^2

Sorry if it's a bit messy. I still haven't gotten the hang of using LaTex yet >_<

 

[Hootenanny]

First one is spot on.

Is this what the second one is meant to look like?

$$y= \left( \frac{t^{2} - 6t}{t^{2} + 6t} \right)^{5}$$

 

[Hollysmoke]

For the 2nd one, yessir!

 

[Hootenanny]

You can simplify it a bit first;

$$y= \left( \frac{t^{2} - 6t}{t^{2} + 6t} \right)^{5} = \frac{(t-6)^5}{(t+6)^5}$$

 

[Hollysmoke]

so should I just use power rule on the top and bottom? or use both quotient and product rule?

 

[Hootenanny]

I would use the quotient rule on this one.

 

[HallsofIvy]

Or write it as
$$(t-6)^5(t+6)^{-5}$$
and use the product rule.

You would not, of course, just differentiate numerator and denominator separately!