# Derivative of inverse tangent function

by biochem850
Tags: derivative, function, inverse, tangent
 P: 46 1. The problem statement, all variables and given/known data Find derivative of tan$^{-1}$($\frac{3sinx}{4+5cosx}$) 2. Relevant equations deriviative of tan$^{-1}$=$\frac{U'}{1+U^{2}}$ 3. The attempt at a solution I found U'= $\frac{12cosx+15}{(4+5cosx)^{2}}$ 1+U$^{2}$=1+$\frac{9sin^{2}x}{(4+5cosx)^{2}}$ I think my components are correct but my answer is still incorrect. Are these basic components even correct (if so I can proceed on my own from here)? 1. The problem statement, all variables and given/known data 2. Relevant equations 3. The attempt at a solution 1. The problem statement, all variables and given/known data 2. Relevant equations 3. The attempt at a solution 1. The problem statement, all variables and given/known data 2. Relevant equations 3. The attempt at a solution 1. The problem statement, all variables and given/known data 2. Relevant equations 3. The attempt at a solution 1. The problem statement, all variables and given/known data 2. Relevant equations 3. The attempt at a solution 1. The problem statement, all variables and given/known data 2. Relevant equations 3. The attempt at a solution 1. The problem statement, all variables and given/known data 2. Relevant equations 3. The attempt at a solution 1. The problem statement, all variables and given/known data 2. Relevant equations 3. The attempt at a solution
 Sci Advisor HW Helper Thanks P: 26,148 hi biochem850! looks ok so far
 P: 46 How do you then simplify: $\frac{U'}{1+U^{2}}$? I've tried to simplify this but with no luck.
 Sci Advisor HW Helper Thanks P: 26,148 hi biochem850! (just got up ) well, the (4 + 5cosx)2 should cancel and disappear … show us your full calculations, and then we'll know how to help!
 P: 46 $\frac{12cosx+15}{(4+5cosx)^{2}}$*1+$\frac{(4+5cosx)^{2}}{9sin^{2}x}$= $\frac{12cosx+15(9sin^{2}x)+(4+5cosx)^{2}(4+5cosx)^{2}}{(4+5cosx)^{2}(9s in^{2}x)}$ I'm not sure this is correct.
HW Helper
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P: 26,148
hmm … let's use tex instead of itex, to make it bigger
 Quote by biochem850 $$\frac{12cosx+15}{(4+5cosx)^{2}} \left(1+\frac{(4+5cosx)^{2}}{9sin^{2}x}\right)$$= $$\frac{12cosx+15}{9sin^{2}x}$$=$$\frac{3(4cosx+5)}{9sin^{2}x}$$
no, that first line is wrong,

the bracket is 1 + 1/U2, it should be 1/(1 + U2)
P: 46
 Quote by tiny-tim hmm … let's use tex instead of itex, to make it bigger … no, that first line is wrong, the bracket is 1 + 1/U2, it should be 1/(1 + U2)
I changed my work just before you posted
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P: 26,148
 Quote by biochem850 I changed my work just before you posted
but you didn't change the bit i said was wrong
P: 46
 Quote by tiny-tim but you didn't change the bit i said was wrong
$\frac{12cosx+15}{(4+5cosx)^{2}}$$\frac{(4+5cosx)^{2}+1}{9sin^{2}x+1}$=

$\frac{12cosx+15}{9sin^{2}x+1}$
I'm not sure this is correct. I really don't understand what you asked me to change.
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P: 26,148
 Quote by biochem850 1+$\frac{9sin^{2}x}{(4+5cosx)^{2}}$ [/SIZE]
… to something with (4 + 5cosx)2 on the bottom
P: 46
 Quote by tiny-tim change your original … … to something with (4 + 5cosx)2 on the bottom

$\frac{(4+5cosx)^{2}+9sin^{2}x}{(4+5cosx)^{2}}$?
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P: 26,148
 Quote by biochem850 $\frac{(4+5cosx)^{2}+9sin^{2}x}{(4+5cosx)^{2}}$?
yup!

now turn that upside-down, and multiply, and the (4 + 5cosx)2 should cancel
 P: 46 $\frac{12cosx+15}{(4+5cosx)^{2}+9sin^{2}x}$ In terms of finding the derivative I know I've found it but this can be simplified further. I don't know if I'm tired or what because I cannot seem to do these very simple problems. This does not bode well for my calculus mark. Would you factor out a 3 in the numerator and then see what will simplify?
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P: 26,148
 Quote by biochem850 Would you factor out a 3 in the numerator and then see what will simplify?
no, and i'd be very surprised if this simplifies

(except that you could get rid of the sin2 by using cos2 + sin2 = 1 )
get some sleep!
P: 46
 Quote by tiny-tim no, and i'd be very surprised if this simplifies (except that you could get rid of the sin2 by using cos2 + sin2 = 1 ) get some sleep!
Through some weird algebraic manipulation I simplified down to $\frac{3}{5+4cosx}$. I beleive this is correct. I'm going to sleep.

Thanks so much!

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