Find d y d x , if x^ 2 3 +y^ 2 3 =a^ 2 3 . - Vidyadip

Question

Find $\frac{d y}{d x},$ if $x^{\frac{2}{3}}+y^{\frac{2}{3}}=a^{\frac{2}{3}}$.

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Answer

Let $x = a \cos^3 \theta, y = a \sin^3\theta$
Then $\frac{d x}{d \theta} = -3a \cos^2 \theta \sin \theta$
and $\frac{d y}{d \theta} = 3a \sin^2 \theta \cos \theta$
Therefore$, \frac{d y}{d x}=\frac{\frac{d y}{d x}}{\frac{d x}{d \theta}}=\frac{3 a \sin ^{2} \theta \cos \theta}{-3 a \cos ^{2} \theta \sin \theta}=-\tan \theta=-\sqrt[3]{\frac{y}{x}}$

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