need calc. help, please

[Guest]

I'm having trouble with the following 2 questions:


1.
Given (x^3)(y)= -y^2 (x), find both dy/dx (treating y as a function of x) and dx/dy ( treating x as a function of y). How are dy/dx and dx/dy related?



2.
Find d/dx ((x^3-2)/x) by using the quotient rule, by using the product rule, and by power rule. Show that your answers are equivalent.

 

[skeeter]

implicit differentiation, correct?

d/dx[x^3*y = -y^2*x]
x^3*(dy/dx) + y*3x^2 = -y^2*1 + x*(-2y)*(dy/dx)
x^3*(dy/dx) + 2xy*(dy/dx) = -3x^2*y - y^2
(dy/dx)(x^3 + 2xy) = -(3x^2*y + y^2)
dy/dx = -(3x^2*y + y^2)/(x^3 + 2xy)

now ...
d/dy[x^3*y = -y^2*x]
x^3*1 + y*3x^2*(dx/dy) = -y^2*(dx/dy) + x*(-2y)

complete the process by solving for dx/dy and then compare with dy/dx.

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use the quotient rule for (x^3 - 2)/x
use the product rule for (x^3 - 2)*x^(-1)
use the general power rule for x^2 - (2/x) = x^2 - 2x^(-1)