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Implicit Differentiation

Implict funct...

Question

Implict function $x^{3} + y^{3} = 3 a x y$

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Solution

$x^{3} + y^{3} = a x y$

By differentiating,

$3 x^{2} + 3 y^{2} \frac{d y}{d x} = a [x \frac{d y}{d x} + y (1)]$

$3 x^{2} + 3 y^{2} \frac{d y}{d x} = a x \frac{d y}{d x} + y a$

$\frac{d y}{d x} (3 y^{2} - a x) = a y - 3 x^{2}$

$\frac{d y}{d x} = \frac{a y - 3 x^{2}}{3 y^{2} - a x}$

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x^{3} + y^{3} = 3 a x y

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Q. The symmetry in curve

x^{3} + y^{3} = 3 a x y

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then prove that

\frac{d^{2} y}{d x^{2}} = \frac{2 a^{2} x y}{{(a x - y^{2})}^{3}}

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