$\text { Find } \frac { d y } { d x } \text { by implicit differentiation given that } 2 x y = 9 \text {. }$
A) $\frac { d y } { d x } = - \frac { 9 y } { x }$
B) $\frac { d y } { d x } = - 9 x y$
C) $\frac { d y } { d x } = - x y$
D) $\frac { d y } { d x } = 9 x y$
E) $\frac { d y } { d x } = - \frac { y } { x }$

Question

$\text { Find } \frac { d y } { d x } \text { by implicit differentiation given that } 2 x y = 9 \text {. }$ 
A) $\frac { d y } { d x } = - \frac { 9 y } { x }$
B) $\frac { d y } { d x } = - 9 x y$
C) $\frac { d y } { d x } = - x y$
D) $\frac { d y } { d x } = 9 x y$
E) $\frac { d y } { d x } = - \frac { y } { x }$

Quizplus · Accepted Answer

We use implicit differentiation to differentiate both sides of the equation with respect to $x$ and apply the product rule on the left-hand side: $$2xy=9 \implies \frac{d}{dx}(2xy)=\frac{d}{dx}(9) \implies 2\frac{dy}{dx} \cdot x + 2y=0.$$ Solving for $\frac{dy}{dx}$, we get $$\frac{dy}{dx}=-\frac{y}{x}.$$ Thus, the answer is $\boxed{	extbf{(E)}}.$

Find dydx by implicit differentiation given that 2xy=9. \text { Find } \frac { d y } { d x } \text { by implicit differentiation given that } 2 x y = 9 \text {. } Find dxdy​ by implicit differentiation given that 2xy=9.

$\text { Find } \frac { d y } { d x } \text { by implicit differentiation given that } 2 x y = 9 \text {. }$