Find $\frac { d y } { d x }$ for the equation $\sqrt { x y } = x - 20 y$ by implicit differentiation and evaluate the derivative at the point $( 100,4 )$ .
A) $- \frac { 1 } { 25 }$
B) $\frac { 1 } { 25 }$
C) $\frac { 3 } { 25 }$
D) $- \frac { 3 } { 25 }$
E)0

Question

Quizplus · Accepted Answer

Using the product rule, we have:$$\frac{d}{dx}\left(\sqrt{xy}\right)=\frac{d}{dx}\left(x-20y\right)$$$$\frac{1}{2\sqrt{xy}}\left(\frac{d}{dx}(xy)\right)=1-20\frac{dy}{dx}$$$$\frac{1}{2\sqrt{xy}}\left(x\frac{dy}{dx}+y\right)=1-20\frac{dy}{dx}$$$$\frac{x}{2\sqrt{xy}}\frac{dy}{dx}+\frac{y}{2\sqrt{xy}}=1-20\frac{dy}{dx}$$$$\left(\frac{x}{2\sqrt{xy}}+20\right)\frac{dy}{dx}=1-\frac{y}{2\sqrt{xy}}$$$$\frac{dy}{dx}=\frac{1-\frac{y}{2\sqrt{xy}}}{\frac{x}{2\sqrt{xy}}+20}$$At the point $(100,4)$, we have:$$\frac{dy}{dx}=\frac{1-\frac{4}{2\sqrt{100\cdot4}}}{\frac{100}{2\sqrt{100\cdot4}}+20}=\frac{1-\frac{1}{10}}{\frac{100}{40}+20}=\frac{\frac{9}{10}}{25+20}=\frac{\frac{9}{10}}{45}=\frac{1}{25}$$Therefore, the correct answer is B.

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