determine dz/dx and dz/dy using implicit differentiation in terms of x, y, z. Given ln(2x^2 + y - z^3) = x?

Answer 1

When differentiating with respect to x, treat y as a constant, and vice versa. Thus,
[1/(2x^2 + y - z^3)]*[4x - (3z^2)*(dz/dx)] = 1; now solve for dz/dx.

For y, we have [1/(2x^2 + y - z^3)]*[1 -(3z^2)*(dz/dy)] = 0. Again, solve for dz/dy.

Answer 2

ln(2x^2 + y - z^3) = x
2x^2 + y - z^3=e^x
z^3=2x^2 + y -e^x
dz/dx=(4x-e^x)/(3z^2)
dz/dy=1/(3z^2)

Answer 3

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