Hi Charles.

With implicit differentiation, you just have to remember that y is a function of x, so if the derivative of y is y′, then the derivative of y² is 2y y′ (I used the chain rule).

So the derivative of y² + x/y + 4x² - 3 (with respect to x, which is what we usually use) is:

Remember that y′ is quite distinct from y, so when you simplify it you should be careful not to combine them improperly.

To actually finish the question, you generally need to solve for y′, but you didn't give us an equal sign to finish the problem (unless perhaps the - 3 should have been = 3). If you meant "equals" 3, then the equation becomes

and you just solve for y′ as in any normal equation.

Hope this helps,
Stephen La Rocque.