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Chapter 2: Problem 10

Find \(d y / d x\) by implicit differentiation. $$ \sqrt{x y}=x-2 y $$

Short Answer

Expert verified
\(\frac{dy}{dx} = \frac{2\sqrt{xy} - y}{x + 4\sqrt{xy}}\)

Step by step solution

01

Apply Implicit Differentiation

Start by applying the derivative to both sides of the equation, with respect to \(x\). For the \(\sqrt{xy}\), we'll have to use the chain rule. The chain rule for this side would look like \(\frac{1}{2 \sqrt{xy}}(\frac{d}{dx}(xy))\). Similarly, two terms on the right side of equation will be differentiated with respect to \(x\), giving us 1 and \(\frac{d}{dx}(2y)\).
02

Execute Chain Rule

Apply the chain rule on the left side as discussed, it becomes \(\frac{1}{2\sqrt{xy}} * (y + x \frac{dy}{dx})\). For the right side, differentiate \(y\) with respect to \(x\) which will result in \(\frac{dy}{dx}\). Therefore, the right side becomes \(1 - 2 \frac{dy}{dx}\). Our equation should now be: \(\frac{1}{2\sqrt{xy}} * (y + x \frac{dy}{dx}) = 1 - 2 \frac{dy}{dx}\).
03

Solve in terms of dy/dx

Finally, we'll have to solve this equation for \(\frac{dy}{dx}\). It's a simple algebraic task from here. Just bring all the dy/dx terms on one side and solve. The final expression for \(\frac{dy}{dx}\) turns out to be \(\frac{dy}{dx} = \frac{2\sqrt{xy} - y}{x + 4\sqrt{xy}}\).

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