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So the question goes like this:

Equation $6xy^{3}+4y-5x^{2}=6$ will define a plot. When we limit the domain and codomain we get a function $y=f(x)$. Find a common expression for the derivative $y'$.

I can not figure out the $y=f(x)$ part of this question. Should I take the $y$ out of the equation to get $f(x)$?

Thanks in advance.

f1tz
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2 Answers2

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In this case $6xy^3+4y-5x^2-6=0$ is a cubic in $y$, which is generally not easy to solve for $y$. If you know some roots, you can factorise, for example $x^3-1=(x-1)(x^2+x+1)$, or you can use Cardano's Formula.

You are asked to find the derivative $y'$, so there is no need to write the function in the form $y=f(x)$, as you can do implicit differentiation.

Indeed we have $$6y^3+18xy^2\frac{dy}{dx}+4\frac{dy}{dx}-10x=0$$

Thus $$\frac{dy}{dx}=\frac{10x-6y^3}{18xy^2+4}=\frac{5x-3y^3}{9xy^2+2}$$

Alessio K
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  • Thank you. I have to study more on implicit differentiation. This helped me to proceed with my task! – f1tz Oct 08 '20 at 12:31
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You are not requested to find the explicit relation $y=f(x)$, you are just asked to find the derivative $$\frac{dy}{dx}$$ which can be done by differentiating the implicit equation.

$$6xy^{3}+4y-5x^{2}=6\to 6y^3+18y^2y'+4y'-10x=0,$$ from which you can draw $y'$.

  • Thank you. I was not sure why I was thinking to find the explicit relation. I have to study more on implicit differentiation. – f1tz Oct 08 '20 at 12:30