Christos
  • Christos
the derivative of tan^3(sqrt(x)) is 3tan^2(sqrt(x))2tan(sqrt(x)) ? I am asking in an attempt to actually find this derivative http://screencast.com/t/1uOEBG1WH
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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katieb
  • katieb
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Christos
  • Christos
@zepdrix
Christos
  • Christos
@dan815 @e.mccormick @Emily778 @Euler271 @Hero @jim_thompson5910 @Luigi0210 @modphysnoob @primeralph
Luigi0210
  • Luigi0210
Chain rule:

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Luigi0210
  • Luigi0210
F(g(x))=F'(g(x))*G'(x)
Christos
  • Christos
So the question is: is my statement in the first post correct , if no why not?
e.mccormick
  • e.mccormick
Now, you have three functions going on there. Do you see that part?
Christos
  • Christos
Honestly I have no idea what's going on , the most difficult thing I ever had to differentiate
Christos
  • Christos
@zepdrix
e.mccormick
  • e.mccormick
Lets rewrite it.\[\tan^3(\sqrt{x})=[tan(x^{\frac{1}{2}})]^3\]Does that help you see three functions?
Christos
  • Christos
hold on
e.mccormick
  • e.mccormick
that last part is just product rule, so not that hard to do once you know the derivative of this part.
Christos
  • Christos
uhm I dunno how you define a function, I cant count it
Christos
  • Christos
tan is a function, x with this power is a function, the third?
e.mccormick
  • e.mccormick
Any time x has something done to it, it can be seen as a new function.
e.mccormick
  • e.mccormick
The whole thing to the 3rd power is the 3rd.
Christos
  • Christos
alright and then?
e.mccormick
  • e.mccormick
So, you ever seen the chain rule applied multiple times?
Christos
  • Christos
just twice
Christos
  • Christos
now it needs three times?
zepdrix
  • zepdrix
The setup will be the same as the last problem ms christos :o \[\large y=\sqrt{x}\tan^3\sqrt{x}\] \[\large y'=\color{royalblue}{(\sqrt{x})'}\tan^3\sqrt{x}+\sqrt{x}\color{royalblue}{(\tan^3\sqrt{x})'}\] Remember how we did that? Product rule setup.
Christos
  • Christos
that gets me (x^(-1/2)tan^3(sqrt(x)))/2 + sqrt(x)3(tan^2(sqrt(x))2tan(sqrt(x)) ?
e.mccormick
  • e.mccormick
\(d/dx~ h(f(n))=h'(f(n))f'(n)\) But what if n=g(x)? Then f'(n) is another chain rule! \(d/dx~ f(g(x))=f'(g(x))g'(x)\) When I put the two together: \(d/dx~ h(f(g(x)))=h'(f(g(x)))f'(g(x))g'(x)\)
zepdrix
  • zepdrix
\[\large y'=\color{orangered}{(\frac{1}{2}x^{-1/2})}\tan^3\sqrt{x}+\sqrt{x}\color{royalblue}{(\tan^3\sqrt{x})'}\]Hmm, your first derivative looks correct, let's see if we can figure out what's going on with the other one.
Christos
  • Christos
ok
zepdrix
  • zepdrix
Let's just look at the blue term for now,\[\large \color{royalblue}{(\tan^3\sqrt x)'}\] So the outermost function is (something) cubed. Applying the power rule and chain rule gives us,\[\large 3(\tan^2\sqrt x)\color{royalblue}{(\tan\sqrt x)'}\]Three came down, giving us a 2 exponent, then we apply the chain rule. Remember the derivative of tangent?\[\large 3(\tan^2\sqrt x)(\sec^2\sqrt x)\color{royalblue}{(\sqrt x)'}\] Are you following my notation ok? The blue terms are ones that we need to take a derivative of. They keep showing up due to the chain rule.
e.mccormick
  • e.mccormick
A lot of people like to start at the inner most function, and work their way out. It is not a bad choice because it works well for multiple chain rules. Or work from outer in. However, as zepdrix points out, there is a problem with part of what you did for a derivative. What is the derivative of tangent? hmm.. he got it as I was typing in my word processor. LOL
zepdrix
  • zepdrix
oh my bad hehe c:
e.mccormick
  • e.mccormick
np. It all works. Not like we are contradicting each other. LOL. I have had that happen.
Christos
  • Christos
I see how it goes now, lets see what I can do about this
e.mccormick
  • e.mccormick
go team go! (Well, a team of 1....)
Christos
  • Christos
(x(-1/2)/2)*tan^3(sqrt(x))+(3sqrt(x)tan^2(sqrt(x))sec^2(sqrt(x))x(-1/2))/2 oh boy
Christos
  • Christos
the second "(" count it as a "^"
Christos
  • Christos
(x^(-1/2)/2)*tan^3(sqrt(x))+(3sqrt(x)tan^2(sqrt(x))sec^2(sqrt(x))x(-1/2))/2
Christos
  • Christos
(x^(-1/2)/2)*tan^3(sqrt(x))+(3sqrt(x)tan^2(sqrt(x))sec^2(sqrt(x))x^(-1/2))/2
zepdrix
  • zepdrix
\[\large y'=\color{orangered}{(\frac{1}{2}x^{-1/2})}\tan^3\sqrt{x}+\sqrt{x}\color{orangered}{(\large 3\tan^2\sqrt x)(\sec^2\sqrt x)(\frac{1}{2}x^{-1/2})}\] yah it looks like you did it correctly. :) should probably be simplified down a smidge though.
Christos
  • Christos
ty bro

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