$\text { Find } \frac { d y } { d x } \text { by implicit differentiation. }$ $x ^ { 2 } + 5 x + 9 x y - y ^ { 2 } = 4$
A) $\frac { d y } { d x } = \frac { 2 x - 5 + 9 y } { 2 y - 9 x }$
B) $\frac { d y } { d x } = \frac { 2 x + 5 + 9 y } { 2 x - 9 y }$
C) $\frac { d y } { d x } = \frac { x + 5 + 9 y } { y - 9 x }$
D) $\frac { d y } { d x } = \frac { 2 x + 5 + 9 y } { 2 y - 9 x }$
E) $\frac { d y } { d x } = \frac { 2 x + 5 - 9 y } { 2 y - 9 x }$

Question

Quizplus · Accepted Answer

We will use implicit differentiation to find the derivative of y with respect to x.

Differentiating both sides with respect to x, we get:

$2x+5+9y+\frac{d}{dx}(9xy)-\frac{d}{dx}(y^2)=0$

Simplifying this expression using the product rule of differentiation, we have:

$2x+5+9y+9x\frac{dy}{dx}+9y-2y\frac{dy}{dx}=0$

Combining like terms and solving for $\frac{dy}{dx}$, we get:

$\frac{dy}{dx}=\frac{2x+5+9y}{2y-9x}$

Therefore, the answer is (D).

Find dydx by implicit differentiation. \text { Find } \frac { d y } { d x } \text { by implicit differentiation. } Find dxdy​ by implicit differentiation.

$\text { Find } \frac { d y } { d x } \text { by implicit differentiation. }$