anonymous
  • anonymous
find the nth order derivative of tan inverse (x/a)?
Precalculus
  • Stacey Warren - Expert brainly.com
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SOLVED
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schrodinger
  • schrodinger
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AravindG
  • AravindG
Try finding the the first 2-3 derivatives and observing the pattern.
zzr0ck3r
  • zzr0ck3r
i find it best to 0(off to the side) next to the function then put a 1 next to the first derivative then put a 2 next to the 2nd derivative . . this helps be build the notation at the end
experimentX
  • experimentX
There you go ... the pattern http://www.wolframalpha.com/input/?i=Table%5BD%5BArcTan%5Bx%2Fa%5D%2C+%7Bx%2C+n%7D%5D%2C+%7Bn%2C+1%2C+10%7D%5D

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AravindG
  • AravindG
^^told ya :)
zzr0ck3r
  • zzr0ck3r
damn you got mad WA skills @experimentX
experimentX
  • experimentX
lol ... no quite, but btw, mathematica solves most of my problems.
zzr0ck3r
  • zzr0ck3r
you even managed to do that without giving away the answer with some f^n(x)...
zzr0ck3r
  • zzr0ck3r
what is your area of expertice?
experimentX
  • experimentX
No, this is not the answer, i don't really know how to find the n-th derivative of it. I would look into Lebnitz or look for some recurrence relation. I am just an old student.
ash2326
  • ash2326
@madrockz do you know the first derivative of this? \[\tan^{-1} \frac{x}{a}\]
anonymous
  • anonymous
yes i know it
ash2326
  • ash2326
okay, so first step is to write that down. Let's try to see if we can get a pattern
experimentX
  • experimentX
also Taylor series of arctan called gregory series ,,, this is quite nice sries http://www.wolframalpha.com/input/?i=Expand+Arctan%28x%29
anonymous
  • anonymous
y1=a^2/(a^2+x^2)
ash2326
  • ash2326
you'll also get a 1/a \[y'=\frac {a^2}{x^2+a^2}\times \frac 1 a\]
anonymous
  • anonymous
ohh yaa...u r right...i missed it but after dat?
ash2326
  • ash2326
let's take the second derivative \[y''=-1\times\frac {a}{(x^2+a^2)^2}\times {2x}\] do you follow?
anonymous
  • anonymous
ok i got it...but wat about the main thing...nth derivative?
ash2326
  • ash2326
Now we'll take the third and then fourth derivative. I believe that a pattern should appear Could you check this in your notebook?
anonymous
  • anonymous
should i give u the 3rd nd 4th derivative?
ash2326
  • ash2326
yes, post them here. Try to use the equation editor, it'll help me understand better
experimentX
  • experimentX
for simplicity's sake put x/a = y df/dx = df/dy.dy/dx = 1/a df/dy
experimentX
  • experimentX
or just put a=1, do you have your final answer in closed for or as infinite series?
anonymous
  • anonymous
\[y3=-[8ax^2/(z^2+a^2)^3 +2a/x^2+a^2)^2]\]
anonymous
  • anonymous
in closed i suppose
experimentX
  • experimentX
It seems that there is no nice closed representation http://functions.wolfram.com/ElementaryFunctions/ArcTan/20/02/
anonymous
  • anonymous
GO THROUGH THIS PDF U WILL GET UR ANSWER....THERE IIS GENERALISED FORM OF YOUR GIVEN QUESTION. SEE POINT NO 2
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