Question

Find dy/dx by implicit differentiation:
x^4(x+y)=y^2(3x-y)

Answer 1

take the derivative of both sides with respect to x, where y is the function y(x)

Answer 2

Im a little confused as to, do I chain rule it? Or product rule it?

Answer 3

You can use both rules to solve it.

x^5+x^4*y=3x*y^2-y^3

To take the derivative of say 3 x y^2, first use the product rule:
d/dx (3x y^2)=3*y^2+3x*d/dx (y^2).
Next use the chain rule:
d/dx (y(x)^2) = 2y(x)*y'(x)
Putting it all together:
d/dx (3x y^2)=3*y^2+3x*2y*y'.