Do I understand this correctly? (Implicit differentiation)

[spice bison]

x^2y uses the power rule, but since y is like unknown, the derivatives of y (which is basically just a function f(x)), so dy/dx (meaning like an unknown variable), so that's why x^2y did not simplify. Also, y^4 uses the chain rule because of the same reason as the first one (y is a function).

[steady fern]

implicit differentiation comes from the chain rule, imagine (2x)^2, differentiating that with the chain rule, youd first multiply by the power and reduce it by 1, then multiply by the inner derivative right? imagine y^4 as (y)^4. do the chain rule, youd multiply by the power, reduce it by one and multiply by the inner functions derivative which here is dy/dx. the deriative of yx^2 arises from the product rule and implicitly differentiating y when necessary

the math in the photo is correct, yes

[spice bison]

But is it correct that y is a function? and dy/dx is basically y prime? Also, is it correct that dy/dx is unknown; therefore, it cannot be simplified in f(g(x)) * g'(x) (chain rule)?

[steady fern]

y is a function of x. dy/dx = y'. dy/dx is currently known until factored and solved

[spice bison]

I see, I guess my understanding is correct, thanks

[steady fern]

you can factor out dy/dx by going dy/dx(4y^3-x^2)=2xy-6x

and then divide by (4y^3-x^2)

[spice bison]

yeah, I just did some khan question after figuring it out 10 minutes ago.

aight, I think it's done, thanks again

.solved