# What is the first derivative of #(y)*(sqrt(x)) - (x)*(sqrt(y)) = 16#?

using the product rule:

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To find the first derivative of the equation ( y\sqrt{x} - x\sqrt{y} = 16 ), differentiate both sides of the equation implicitly with respect to ( x ).

Using implicit differentiation and the chain rule, the first derivative is:

[ \frac{dy}{dx}\sqrt{x} + \frac{y}{2\sqrt{x}} - \sqrt{y} - x\left(\frac{1}{2\sqrt{y}}\frac{dy}{dx}\right) = 0 ]

Simplify the expression to solve for ( \frac{dy}{dx} ), the first derivative:

[ \frac{dy}{dx}\sqrt{x} - x\left(\frac{1}{2\sqrt{y}}\frac{dy}{dx}\right) = \sqrt{y} - \frac{y}{2\sqrt{x}} ]

[ \frac{dy}{dx}\left(\sqrt{x} - \frac{x}{2\sqrt{y}}\right) = \sqrt{y} - \frac{y}{2\sqrt{x}} ]

[ \frac{dy}{dx} = \frac{\sqrt{y} - \frac{y}{2\sqrt{x}}}{\sqrt{x} - \frac{x}{2\sqrt{y}}} ]

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