As the first step, we will differentiate once, and apply the product rule:
d/dx[x^3]*y^3 + d/dx[y^3]*x^3 = d/dx[8]ddx[x3]⋅y3+ddx[y3]⋅x3=ddx[8]
For y^3y3, remember to use the chain rule. Simplifying yields:
3x^2y^3 + 3y^2x^3dy/dx = 03x2y3+3y2x3dydx=0
Now, we will solve for dy/dxdydx:
dy/dx = -(3x^2y^3)/(3y^2x^3)dydx=−3x2y33y2x3
We can cancel off the 3, an x^2x2, and a y^2y2, which will yield:
dy/dx = -y/xdydx=−yx
Now, differentiate once again. We will apply the quotient rule:
(d^2y)/(dx^2) = -(x*dy/dx - y*1)/x^2d2ydx2=−x⋅dydx−y⋅1x2
Looking back at the previous equation for dy/dxdydx, we can substitute into our equation for the second derivative to get it in terms of only xx and yy:
(d^2y)/(dx^2) = -(x*(-y/x) - y*1)/x^2d2ydx2=−x⋅(−yx)−y⋅1x2
Simplifying yields:
(d^2y)/(dx^2) = (2y)/x^2d2ydx2=2yx2