Anyone good in Calculus here? Derivatives?

superman22x

Golden Master
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So I am totally lost... I need some help. Thought I might ask here...
Problem:
Find the derivative of the following:

(1+cscX)/(1-cscX)

Answer is:

(-2cscXcotX)/((1-cscX)^2)

Can someone explain how I get to that answer? I am lost... That's just one problem. But I don't understand how to get there.

The d/dX of cscX = -cscXcotX
 
Might be a little hard to explain through the internet, but I'll give it a shot. Are you solving by definition of derivative (limits) or using rules?
 
haha, i was about to post saying that im pretty sure JogaBonito1502 would be able to help you out.
but it looks like he beat me to the punch.
 
Oh geez. This is going to be a long one. Here I go:

There's 2 ways to do by limits. As x -> c and as delta x (for lack of sign I'll use #) -> x. So with signs as #x -> x. I'll do the #x method because I find it easier to simplify. It can be even easier once you start using Power, Product, Quotient, and Chain rules.

So:

lim ........................... 1 + csc(#X + X) ................... 1 + cscX
#X->0 ...................... ---------------- ..... - ........ -----------
................................1 - csc(#X + X) ...................... 1 - cscX
...............................-------------------------------------------
...........................................................#X

lim.................................... (1 - cscX)(1 + csc(#X+X)) - (1 - csc(#X+X))(1 + cscX)....................... All that I've done here is common denominators
#X->0.............................. -------------------------------------------------------................................... followed by simplification.
.........................................................#X(1 - csc(#X+X))(1 - cscX)


lim....................................1 + csc(#X+X) - cscX - (cscX)(csc(#X+X)) - 1 - cscX + csc(#X+X) + (cscX)(csc(#X+X))
#X->0..............................------------------------------------------------------------------------------------------................Distribute.
.........................................................................#X(1 - csc(#X+X))(1 - cscX)......................................................I've already taken care of distributing the -1.

lim..........................................................2csc(#X+X) - 2cscX
#X->0................................................----------------------------.............Simplify
.........................................................#X(1 - csc(#X+X))(1 - cscX)

lim....................................................2(csc(#X+X) - cscX)
#X->0............................................-----------------------.............Simplify more (it's always good in a hard problem like this; I just use rules)
.......................................................#X(1 - csc(#X+X))(1-cscX)


Taken out of context to simplify:

...................................1...............................................1...........................
...................------------------------......-........--------------------------........ Rule for simplifying sin(A+B)
...................sin#X cosX + cos#X sinX..............................sinX........................

............................sinx - sin#XcosX - cos#XsinX
...................--------------------------------------
......................sinXsin#XcosX + cos#X(sinX)^2

Alright dude, screw this we're doing rules, as proving trig stuff is really nasty. Even my calc teacher said so. After tons of simplification you should get:

lim....................................................................2cscX
#X->.............................................................-----------.........................The #X on the side needs to be cancelled off so you get:
.........................................................#X(1 - csc(#X+X))(1 - cscX)

lim....................................................................2cscX
#X->.............................................................-----------.........................Plug 0 for #X
........................................................ (1 - csc(#X+X))(1 - cscX)

lim....................................................................2cscX
#X->.............................................................-----------.........................And there's your answer.
..............................................................(1 - cscX)(1 - cscX)
 
Ok, might be rules.

Thinking about it again. The formal definition is F(X+H)-F(X)
-------------
H
Correct? But with the qoutent rule, I would take u1v - uv1
-----------
v^2
u= 1+cscX
v= 1-cscX
1= take derivative

maybe that's it... Or is there simplifying I can do? I know there are rules I can apply to csc sec and cot, but I cant remember them... secx = 1/sinx correct? cotx = 1/tanx? and cscx = 1/cosx
 
K man, I started with definition I'll keep going. I'll show rule after k?


Through rules:

(1 + cscX)
----------
(1 - cscX)


(1 + cscX)(1 - cscX)^-1

-cscXcotX(1-cscX)^-1 + (1+cscX)(-1)(-cscXcotX)^-2

-cscXcotX.........1 + cscX
-----------..+..-----------
1 - cscX...........cscXcotX

Do your simplification, etc. I used product rule because I hate quotient rule. Here's quotient:

d.......................................'(1 + cscX)(1 - cscX) - (1 + cscX)'(1 - cscX)
--.... (1 + cscX)/(1 - cscX) = -------------------------------------------
dx...........................................................(1 - cscX)^2

General Form:

d...............('a)(b) - (a)('b)
-- (a/b) = ----------------
dx....................(b)^2

edit: Remind me to never again attempt to help with calculus over the internet. Also, go to calcchat.com and they give you answers to odds questions along with explanations.

edit2: Sorry I missed your explanation of quotient rule. I was too busy trying to get this. You didn't need my help after all =o. Had it in you from the beginning.
 
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