Question

find the nth order derivative of tan inverse (x/a)?

Answer 1

Try finding the the first 2-3 derivatives and observing the pattern.

Answer 2

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

Answer 3

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

Answer 4

^^told ya :)

Answer 5

damn you got mad WA skills @experimentX

Answer 6

lol ... no quite, but btw, mathematica solves most of my problems.

Answer 7

you even managed to do that without giving away the answer with some f^n(x)...

Answer 8

what is your area of expertice?

Answer 9

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.

Answer 10

@madrockz do you know the first derivative of this?
\[\tan^{-1} \frac{x}{a}\]

Answer 11

yes i know it

Answer 12

okay, so first step is to write that down. Let's try to see if we can get a pattern

Answer 13

also Taylor series of arctan called gregory series ,,, this is quite nice sries
http://www.wolframalpha.com/input/?i=Expand+Arctan%28x%29

Answer 14

y1=a^2/(a^2+x^2)

Answer 15

you'll also get a 1/a
\[y'=\frac {a^2}{x^2+a^2}\times \frac 1 a\]

Answer 16

ohh yaa...u r right...i missed it but after dat?

Answer 17

let's take the second derivative
\[y''=-1\times\frac {a}{(x^2+a^2)^2}\times {2x}\]
do you follow?

Answer 18

ok i got it...but wat about the main thing...nth derivative?

Answer 19

Now we'll take the third and then fourth derivative. I believe that a pattern should appear

Could you check this in your notebook?

Answer 20

should i give u the 3rd nd 4th derivative?

Answer 21

yes, post them here.
Try to use the equation editor, it'll help me understand better

Answer 22

for simplicity's sake put x/a = y
df/dx = df/dy.dy/dx = 1/a df/dy

Answer 23

or just put a=1, do you have your final answer in closed for or as infinite series?

Answer 24

\[y3=-[8ax^2/(z^2+a^2)^3 +2a/x^2+a^2)^2]\]

Answer 25

in closed i suppose

Answer 26

It seems that there is no nice closed representation
http://functions.wolfram.com/ElementaryFunctions/ArcTan/20/02/

Answer 27

GO THROUGH THIS PDF U WILL GET UR ANSWER....THERE IIS GENERALISED FORM OF YOUR GIVEN QUESTION. SEE POINT NO 2