monroe17
Find dy/dx
step by step? please? :)



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monroe17
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\[y=ln(\frac{ x^4e^{3x} }{ \sqrt{2x^51} })\]

dpaInc
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yuck!!!

monroe17
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I know ;(

dpaInc
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i think it would be better to do logarithmic differentiation...

dpaInc
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\(\large y=ln(\frac{x^4e^{3x}}{(2x5)^{\frac{1}{2}}}) = 4lnx + 3x  \frac{1}{2}(2x^51)\)
simplifying y that way should be much simpler...

dpaInc
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i guess that's not so hard afterall....

monroe17
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wait wait.. how'd you get 4ln(x)+3x1/2(2x^51)

dpaInc
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use the logaritm properties...

monroe17
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I need to look those up ;/ and review them.

dpaInc
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\(\large ln(AB)=lnA+lnB \);
\(\large ln(A/B)=lnAlnB \);
\(\large ln(A^B)=BlnA \);
hmm... i think that's all of 'em...

monroe17
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Awe, well thank you. Okay so now back to what you got.. Let me look over that real quick

dpaInc
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yeah... should be easier now....

monroe17
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Okay so I see you used the second one.. But, I'm confused how you got those values...
ln(A)=4lnx+3x
when A= x^4e^{3x}

monroe17
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is it..ln(x^4)=4lnx

dpaInc
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\(\large y=ln(\frac{x^4e^{3x}}{(2x5)^{\frac{1}{2}}}) =ln(x^4e^{3x})ln(2x5)^{1/2} \)

dpaInc
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that's the first step...

dpaInc
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now....
\(\large ln(x^4e^{3x})=lnx^4+lne^{3x}=4lnx+3x \)
do u undestand this part?

dpaInc
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ooo... sorry my bad... the second part is wrong....

monroe17
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ohhh... ln(A^b)=b*ln(A)
ln(x^4)=4*ln(x)
and e&ln cancel out leaving 3x

dpaInc
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\(\large ln(2x5)^{1/2}=\frac{1}{2}ln(2x5) \)
that's what it should have been...

dpaInc
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and yes (to your last post)

dpaInc
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lemme rewrite what i shouldve written....

dpaInc
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\(\large y=ln(\frac{x^4e^{3x}}{(2x5)^{\frac{1}{2}}}) = 4lnx + 3x  \frac{1}{2}\color {red}{ln}(2x^51) \)

monroe17
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\[\frac{ 4 }{ x }+3..\]

dpaInc
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yep... keep going...

monroe17
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I need to do chain rule?

dpaInc
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yes, chain rule: \(\large [lny]'=\frac{y'}{y} \)

dpaInc
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so \(\large [ln(2x^51)]'=\frac{[2x^51]'}{2x^51} \)
simplify....

monroe17
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\[\frac{ 4 }{ x }+3\frac{ 1 }{ 2 }\frac{ 10x^4 }{ 2x^51 } = \frac{ 4 }{ x }+3\frac{ 5x^4 }{ 2x^51 }\]

dpaInc
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haha... looks good..:)

monroe17
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why haha? did I do something wrong?

dpaInc
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no... i was getting the wrong answers until at the last moment when u posted ur answer... it confirmed my last answer...

monroe17
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oh lol :) you just discouraged me! >;[
no im jk ;) thank you!

dpaInc
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oh wait...

dpaInc
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haha... just kidding....
yw... :)

monroe17
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haha! jerk ;D

dpaInc
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:)

dpaInc
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i forgot...
nice work on your part... way to stick with the problem....

monroe17
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what you mean? I always finish ;D