Finding Points on Curve where Derivative Doesn't Exist

[AXidenT]

Homework Statement

Find dy/dx on the curve

1/3*y^3 + y*sin(x) = 1 - x^6

At which points on the curve does this derivative does not exist? Find the slope of the line tangent to the curve at the point (0,3^(1/3)).

Homework Equations

dy/dx = - Fx/Fy

The Attempt at a Solution

I solved dy/dx implicitly using the above formula to get:

dy/dx = - (y*cos(x) + 6x^5) / (y^2 + sin(x))

To find the points where dy/dx doesn't exist, I let:

y^2 + sin(x) = 0

I've then tried rearranging the above expression in terms of y and x and substituing into the original curve equation, but can't complete the algebra by hand. Wolfram returned multiple answers depending how it's done. Any help?

Thank you! :)

EDIT: And for the last bit, I would just substitute those values for x and y into dy/dx which gives me something about -0.6 from memory?

 

[mfb]

If sin(x)=-y^2,
1/3*y^3 - y^3 = 1 - x^6
-2/3 y^3 = 1-x^6
y=...
This can be put in y^2 + sin(x) = 0 again to get a (messy) function of x only.
In the same way, you can solve for x and get a function of y only.
I don't think you are supposed to give analytic solutions to those equations.

EDIT: And for the last bit, I would just substitute those values for x and y into dy/dx which gives me something about -0.6 from memory?

The values where the derivative does not exist? That should not give a number for the derivative.

 

[AXidenT]

So I can't get the values seen here:

http://www.wolframalpha.com/input/?i=1/3*y^3+y*sin(x)=1-x^6,+y^2+sinx=0

by hand? The algebra seems to reach a dead end every way I try it. How would I state my answer for the points of the curve that dy/dx is undefined then?

And the values for the part of the question asking: "Find the slope of the line tangent to the curve at the point (0,3^(1/3))."

I got-0.69, which looks right on the graph at that point?

Thank you! :)

EDIT: Also for showing the implicit derivative I should just be able to use dy/dx = -Fx/Fy, without showing how to arrive at that formula? Like I shouldn't have to do it with the chain rule?

 

[mfb]

How would I state my answer for the points of the curve that dy/dx is undefined then?

I would describe them as "the two solution of this set of equations".

And the values for the part of the question asking: "Find the slope of the line tangent to the curve at the point (0,3^(1/3))."

I got-0.69, which looks right on the graph at that point?

That is not a point, but if you mean x=0,3^(1/3), then it looks reasonable.

EDIT: Also for showing the implicit derivative I should just be able to use dy/dx = -Fx/Fy, without showing how to arrive at that formula? Like I shouldn't have to do it with the chain rule?

Should be fine.