Question

Find d2y/dx2 by implicit differentiation, given x3 y3 = 1.

Answer 1

To find the second derivative d2y/dx2, we first implicitly differentiate the equation x3 y3 = 1 with respect to x to find dy/dx and then differentiate dy/dx again with respect to x to get d2y/dx2 which results in (y2 + xy)/(x3).

Step-by-step explanation:

To find the second derivative d2y/dx2 through implicit differentiation for the equation x3 y3 = 1, we first differentiate both sides of the equation with respect to x, treating y as a function of x. This gives us:

1. 3x2y3 + 3x3y2(dy/dx) = 0.
2. Now, solving for dy/dx, we get:
3. dy/dx = -y/x, after simplifying.

Next, we differentiate dy/dx with respect to x again to find d2y/dx2.

d2y/dx2 = d/dx (-y/x)
= -(x(dy/dx) - y)/(x2)
= -((-y2/x) - y)/(x2) after substituting dy/dx
= -(-y2 - yx)/(x3)
= (y2 + xy)/(x3) after simplifying.

This gives us the second derivative of y with respect to x, d2y/dx2, as (y2 + xy)/(x3).