find tangent line x^3+y^3=y+21 at (3,-2)

I got y=-27/11x +59/11 but it's saying its not right, and I cant see the error in my ways

Question

find tangent line x^3+y^3=y+21 at (3,-2)

I got y=-27/11x +59/11 but it's saying its not right, and I cant see the error in my ways

anonymous · Answer

oh hmm shouldnt it be 3x^2 +3y^2(dy/dx)=dy/dx and after solving that its the slope?

myininaya · Answer

yep! :)

myininaya · Answer

$$3y^2 y'-y'=-3x^2$$

myininaya · Answer

$$y'(3x^2-1)=-3x^2$$

myininaya · Answer

Now divide both sides by $$3x^2-1$$

zarkon · Answer

$$3y^2-1$$

myininaya · Answer

oops type-0

myininaya · Answer

yes y doesn't usually change to x unless some kind of act of god happens

myininaya · Answer

or myininaya screws up

zarkon · Answer

I like doing it the quick way

$$\frac{dy}{dx}=-\frac{\frac{\partial f}{\partial x}}{\frac{\partial f}{\partial y}}$$

anonymous · Answer

wait what you just typed to me is the slope? cuz thats dy/dx implicit d confuses the hell out of me

myininaya · Answer

just use y' instead

anonymous · Answer

dy/dx is the slope?

anonymous · Answer

y'

myininaya · Answer

y' always looks prettier to me

myininaya · Answer

$$y'=\frac{dy}{dx}$$

anonymous · Answer

cuz the equation I got up top is the tangent line with the slope as y'

myininaya · Answer

wait what is your question? i'm confused

anonymous · Answer

hahaha okay so how to do I get the slope of the tangent line?

myininaya · Answer

$$slope=y'=\frac{dy}{dx}=\frac{-3x^2}{3y^2-1}$$

anonymous · Answer

hmmm thats -27/11 and it isnt giving me the correct answer

anonymous · Answer

the tangent line I posted is it with that slope so idk what i did wrong

myininaya · Answer

$$y'|_{(3,-2)}=\frac{-3(3)^2}{3(-2)^2-1}=\frac{-27}{11}$$
I don't think I understand your question
yes that is the right slope

myininaya · Answer

$$y=\frac{-27}{11}x+b$$
What point do we know that is on this line (3,-2) 
plug this in to find b

anonymous · Answer

okay so if that is my slope then y+2=-27/11(x-3) y= -27/11x +59/11 but it is saying that is incorrect

ash2326 · Answer

@Awstinf try  typing it in standard form
$$27x+11y=\frac{59}{11}$$

ash2326 · Answer

Sorry$$27x+11y=59$$

myininaya · Answer

That is right lol

myininaya · Answer

Ok well that just means you should subtract 2 both sides

myininaya · Answer

$$y=\frac{-27}{11}(x-3)-2$$
but  you already did

myininaya · Answer

so it just says y=?

anonymous · Answer

yeah :\

myininaya · Answer

So it does it say to put in slope intercept form?

anonymous · Answer

yeah, but webwork is kind of screwy. Probably one of the worst systems I've ever dealt with. Wolfram should make something like it so we never have to deal with it ever again lol.

myininaya · Answer

I'm afraid I don't know what to tell you
Your answer is right

anonymous · Answer

thank you for helping me! again!