more implicit differentials

for what values of x does the curve y^2 -x^4 + 2xy -18x^2 = 10 have vertical tangent lines?

Question

more implicit differentials 

for what values of x does the curve y^2 -x^4 + 2xy -18x^2 = 10 have vertical tangent lines?

anonymous · Answer

@ganeshie8 @IrishBoy123 any ideas?

zepdrix · Answer

Horizontal tangent lines when: $\large\rm y'=0$
Vertical tangent lines when: $\large\rm y'=\frac{stuff}{0}$

Have you tried finding your y' yet? :)

anonymous · Answer

yes

anonymous · Answer

i feel i messed up somewhere though so im redoing that right now

anonymous · Answer

im getting 

$$y'= \frac{ 4x(x^2+9) }{ 2(y+x) }$$

zepdrix · Answer

Hmm I think there is another term on top.
Did you forget to product rule again? :)$$\large\rm y^2 -x^4 + \color{orangered}{2xy} -18x^2 = 10$$

anonymous · Answer

no i just simplified from 4x^3 + 36x

zepdrix · Answer

But where is the -2y in the numerator? :o hmm

anonymous · Answer

oh your totally right

anonymous · Answer

argh your so clever :P

zepdrix · Answer

$$\large\rm y^2 -x^4 + 2xy -18x^2 = 10$$Differentiating gives,$$\large\rm 2yy'-4x^3+2y+2xy'-36x=0$$That's your first step ya? :D

zepdrix · Answer

you're*
that's gonna bug me, i had to lol

anonymous · Answer

yeah i even crossed it out and all i guess it just slipped my mind when i was rewriting it on the other side of the equal sign

anonymous · Answer

anyway so its $$y'= \frac{ 4x^3 + 36x + 2y }{ 2(x+y) }$$

zepdrix · Answer

Woops, -2y on top I think ya?

zepdrix · Answer

Anyway, let's just get rid of all the 2's I guess,$$\large\rm y'=\frac{2x^3+18x-y}{x+y}$$That's the only simplification that really cleans it up nicely.

anonymous · Answer

:( yes , okay so now what?

zepdrix · Answer

This derivative function is undefined when the denominator is zero.
(This is also when we're getting vertical tangents.)

zepdrix · Answer

So vertical tangent when the denominator is zero,
$\large\rm x+y=0$

zepdrix · Answer

Overheat again? :) LOL

jhannybean · Answer

|dw:1441787497363:dw| where its undefined? :P

anonymous · Answer

yep :( i need a new computer

zepdrix · Answer

Hmm ya that's a weird answer :o I do something wrong?

anonymous · Answer

anyway one of the answer choices is x = -y so i think that's the answer right?

zepdrix · Answer

yay team \c:/
it just doesn't make a whole lot of sense with the graph of the function :D
I guess I just need to think about it a sec lol

jhannybean · Answer

I think that's about right, it's either $\sf y=-x$ or $\sf x=-y$.

anonymous · Answer

thank you once more :D