use implicit differentiation to find the derivative of $(x^2+y^2)^4=6x^2y\,$?

Question

I made up a question to practice implicit differentiation with the relation $(x^2+y^2)^4=6x^2y$. this is my solution:

Also I am sorry but I don't know how to write the more complex parts of the solution in the question. I would just like to know if I was correct because I do not know how to verify my answer with this type of question, and if anyone has some tips on how to do that it would be much appreciated.

Answer 1

Your solution is correct. And yes, there is a short-cut trick which you can use to verify your answer.

For an implicit equation in $x$ and $y$, put all the terms on left side and assume that to be a function of $x$ and $y$ such that $f(x,y) = 0$.

Now, $$ \dfrac{dy}{dx} = -\dfrac{f_x}{f_y}$$

Here, $f_x$ and $f_y$ are partial derivatives of the function $f$ with respect to $x$ and $y$ respectively.

You can use it directly to save much time.

---

You can also use WolframAlpha to check your answer.

Answer 2

Your solution is correct. As far as I know there in now way to verify your answer, you can just re-do the calculations to be sure. If an expression confuse you, you can also write $y$ as $f(x)$ and use Lagrange's notation $f'(x)$ since you know that you differentiate with respect to $x$.