find (dy/dt) when x= -2, dx/dt = 2, x^3 + y^3 = 19

Question

anonymous · Answer

x^3 + y^3 = 19
3x^2x'  + 3y^2y' = 0
-> y' = - x^2x'/ y^2
At x = -2, x' = 2 : 
y' = - 8/ y^2

anonymous · Answer

Hey Jinnie, no more related rate, today :)

anonymous · Answer

Implicit differentiation. Basically we just go ahead and differentiate both sides, but you must remember that you are differentiating with respect to (t), so when you take the derivative of y and x, you must chain rule.

$$x^3 + y^3 = 19$$

$$d/dt ( x^3 + y^3) = d/dt (19)$$

$$3x^2 x' + 3y^2 y' = 0$$

now lets go ahead and plug in the numbers they gave us

$$3(-2)^2 (-2) + 3y^2 y' = 0$$

$$-24 + 3y^2y' = 0$$

$$3y^2y' = 24$$

$$y' = 24/(3y^2)$$

$$y' = 8/(y^2)$$

Usually you can just leave the y in the answer, but if you don't want to, just go isolate y in your original equation and plug it in.

also remebmer dy/dt is the same as y', just different notation.

anonymous · Answer

thanks both

anonymous · Answer

Nada, Jinnie :)
@Hermeezey, what a terrific explanation. It's going to cost me the whole day to think like you!

anonymous · Answer

lol i appreciate both. but is it -8 or +8

anonymous · Answer

Haha thank you Chlorophyll. I just wanted to be as clear as possible

anonymous · Answer

It's -8, because when we switch to other side !

anonymous · Answer

ok thanks. hey cholorophyll can you refresh my memory on what to do to differentiate when you're given (g of f)' (x)

anonymous · Answer

I always strive for neat and clear, but I'm neither good explainer nor typist :(

anonymous · Answer

Give me the example, Jinnie!

anonymous · Answer

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(g of f) (4)