and evaluate the derivative at the given point.

x^(2/3) + y^(2/3)=5 (8,1)

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- Sep 26th 2009, 01:20 PM #1

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- Sep 26th 2009, 01:30 PM #2
$\displaystyle y^{\frac{2}{3}} = 5 - x^{\frac{2}{3}}$

$\displaystyle y = (5-x^{\frac{2}{3}})^{\frac{3}{2}}$

By the chain rule:

$\displaystyle \frac{dy}{dx} = \frac{3}{2}(5-x^{\frac{2}{3}})^{\frac{1}{2}} \times -\frac{2}{3}x^{-\frac{1}{3}}$

$\displaystyle

=-\frac{\sqrt{5-x^{\frac{2}{3}}}}{\sqrt[3]{x}}$

$\displaystyle f'(8) =-\frac{\sqrt{5-8^{\frac{2}{3}}}}{\sqrt[3]{8}} = -\frac{1}{2}$

- Sep 26th 2009, 01:39 PM #3

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- Sep 26th 2009, 01:50 PM #4