Complex Derivatives !!
if y = ((6x-3)^4)/((3x+4)^(1/3)) what is dy/dx ?

- anonymous

Complex Derivatives !!
if y = ((6x-3)^4)/((3x+4)^(1/3)) what is dy/dx ?

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- anonymous

\[y = \frac{ (6x-3)^4 }{ \sqrt[3]{3x+4} }\]

- anonymous

okay so let me show you what i got and then tell me what iv done wrong

- DanJS

did you move the root to the top as -1/3 power, and apply the product rule

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## More answers

- anonymous

!!! no i was doing the quotient rule like the fool i am!!! very clever!!

- anonymous

okay let me start over

- DanJS

either is good, it isnt too much work

- anonymous

okay so this is what iv got so far
\[(24(6x-3)^3)((3x+4)^{-1/3}) + ((6x-3)^4)(-(3x+4)^{-4/3})\]

- anonymous

quick question : i cant distribute the 24 into the (6x-3) because its cubed right?

- DanJS

no

- anonymous

so i can distribute the 24?

- DanJS

test and see if you forget algebra things...
24(6x - 3)^3 = the distributed value?

- DanJS

use any number for x

- anonymous

okay sorry lost internet connection,

- DanJS

maybe do the quotient rule and see if that is easier

- anonymous

not sure where to go now, do i just bring down the negative exponents?

- anonymous

\[(24(6x-3)^3 + (6x-3)^4)/(-(3x+4)^{4/3}+(3x+4)^{1/3})\]

- anonymous

??

- DanJS

that is the derivative, done..

- DanJS

i am sure they want it all simplified though i assume

- anonymous

yep it seems so , it dosent appear to be any of the answer choices

- DanJS

you can move the - exponents , but you can not do that,
a * b^-1 + c*d^-1 is not same as (a +c) / (b + d)

- DanJS

rather it is just (a/b) + (c/d)

- anonymous

okay don't worry about the simplification it is just tedious work i think iv got it , thank you very much for your help

- DanJS

notice the parenthesis value appears in both terms, for both the different parenthesis,
maybe try some factoring

- DanJS

\[(24(6x-3)^3)((3x+4)^{-1/3}) + ((6x-3)^4)(-(3x+4)^{-4/3})\]

- DanJS

or, put the negative exponents back to the bottom, then you would have to do the common denominator to add the two fractions

- DanJS

\[\frac{ 24(6x-3)^3 }{ (3x+4)^{1/3} } + \frac{ (6x-3)^4 }{ (3x+4)^{4/3}}\]

- DanJS

need to multiply first term by (3x+4)^{3/3} / (3x+4)^{3/3}

- anonymous

you mean 4/3 for your exponent right?

- anonymous

not that it matters much now but we did it completely wrong,
we were suppose to multiply both sides by Ln at the beginning and work from there

- DanJS

yes, i just skipped a couple things, the net effect is what i said
the common denominator to multiply everything by is the product of the two, so you get
(3x+4)^{5/3}
that puts the denominator to 5/3 power, and adds a factor of (3x+4)^{1/3} that can factor from the top two terms,
so overall you are just multiplying the first fraction by (3x+4)^(3/3)

- DanJS

\[\frac{ (3x+4)*24*(6x-3)^2 + (6x-3)^4 }{ (3x+4)^{4/3} }\]

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