find dy/dx if y^3-xsiny+y^2/x=8

To find the derivative of the given function, we'll use the rules of differentiation, namely the power rule, the product rule, the chain rule, and the quotient rule.

The given function is y^3 - x*sin(y) + y^2/x = 8.

Step 1: Differentiate each term separately.

The derivative of y^3 with respect to x is 3y^2*(dy/dx) using the power rule and the chain rule (since y is a function of x).

The derivative of -xsin(y) with respect to x is -sin(y) - xcos(y)*(dy/dx) using the product rule and the chain rule.

The derivative of y^2/x with respect to x is (2y*(dy/dx)*x - y^2)/x^2 using the quotient rule.

The derivative of 8 with respect to x is 0.

Step 2: Combine all the derivatives.

So, the derivative of the given function is 3y^2*(dy/dx) - sin(y) - xcos(y)(dy/dx) + (2y*(dy/dx)*x - y^2)/x^2 = 0.

Step 3: Solve for dy/dx.

Group all the terms with dy/dx on one side and the remaining terms on the other side.

(3y^2 - xcos(y) + 2y/x)(dy/dx) = sin(y) + y^2/x^2.

Finally, dy/dx = [sin(y) + y^2/x^2] / [3y^2 - x*cos(y) + 2y/x].

This is the derivative of the given function.