Implicit Differentiation
Step-by-step explanation:
Given:
[tex]x^{\frac{2}{3}} -y^{\frac{2}{3}} = 6\\[/tex]
Recall:
[tex]x^{\frac{2}{3}} -y^{\frac{2}{3}} = 6 \\ \frac{\text{d}}{\text{d}x}(x^{\frac{2}{3}} -y^{\frac{2}{3}}) = \frac{\text{d}}{\text{d}x}(6) \\ \frac{\text{d}}{\text{d}x}(x^{\frac{2}{3}}) -\frac{\text{d}}{\text{d}x}(y^{\frac{2}{3}}) = 0 \\ \frac{2}{3}x^{\frac{2}{3} -1} -\frac{2}{3}y^{\frac{2}{3} -1}\cdot \frac{\text{d}y}{\text{d}x} = 0 \\ \frac{2}{3}x^{-\frac{1}{3}} - \frac{2}{3}y^{-\frac{1}{3}} \cdot \frac{\text{d}y}{\text{d}x} = 0 \\ \frac{2}{3\sqrt[3]{x}} -\frac{2}{3\sqrt[3]{y}}\cdot \frac{\text{d}y}{\text{d}x} = 0 \\ -\frac{2}{3\sqrt[3]{y}}\cdot \frac{\text{d}y}{\text{d}x} = -\frac{2}{3\sqrt[3]{x}} \\ -2 \cdot \frac{\text{d}y}{\text{d}x} = -\frac{6\sqrt[3]{y}}{3\sqrt[3]{x}} \\ \frac{\text{d}y}{\text{d}x} = -\frac{6\sqrt[3]{y}}{-6\sqrt[3]{x}} \\ \frac{\text{d}y}{\text{d}x} = \frac{\sqrt[3]{y}}{\sqrt[3]{x}}[/tex]
