Question

differential of (dy/dx)=y/x

Answer 1

If you want the second derivative, d^2y/dx^2, use the chain rule (with y treated as a function of x) together with the rule for differentiating a fraction or product.

d^2y/dx^2 = -y/x^2 + (1/x)(dy/dx)
= -y/x^2 +y/x^2
= 0

How can that be? Here's why.

The only function y(x) for which the slope dy/dx is always y/x, is a straight line through the origin, and has no curvature. that means d^2y/dx^2 must be zero.

Cute problem!