Given that $3{{x}^{2}}{{y}^{2}}=4{{x}^{2}}-4xy$, find an expression of \(\frac{{\mathrm{d}y}}

Question

Given that $3{{x}^{2}}{{y}^{2}}=4{{x}^{2}}-4xy$, find an expression of $\frac{{\mathrm{d}y}}{\mathrm{d}x}$.

Accepted Answer

For $x\ne0$, $\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{2y}{x(3xy+2)}=\dfrac{4x-2y-3xy^2}{3x^2y+2x}$. The curved branch through $(0,0)$ has derivative $1$; the vertical component $x=0$ has no finite $\dfrac{\mathrm dy}{\mathrm dx}$.

Techniques of Differentiation - Implicit Differentiation

Techniques of Differentiation - Implicit Differentiation

Solution:

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