Math  /  Calculus

QuestionConsider the following equation. 6x2y2=46 x^{2}-y^{2}=4 (a) Find yy^{\prime} by implicit differentiation. y=y^{\prime}= \square (b) Solve the equation explicitly for yy and differentiate to get yy^{\prime} in terms of xx. y=±y^{\prime}= \pm \square Need Help? Read It Watch It Submit Answer 14. [-/4.76 Points] DETAILS MY NOTES SCALCET9 3.5.009.
Find dydx\frac{d y}{d x} by implicit differentiation. x2x+y=y2+2dydx=\begin{array}{l} \frac{x^{2}}{x+y}=y^{2}+2 \\ \frac{d y}{d x}=\square \end{array}

Studdy Solution
Solve for dydx\frac{dy}{dx}:
Rearrange and simplify to solve for dydx\frac{dy}{dx}:
(2x(x+y)x2)x2dydx=2y(x+y)2dydx (2x(x+y) - x^2) - x^2 \frac{dy}{dx} = 2y(x+y)^2 \frac{dy}{dx}
Combine terms involving dydx\frac{dy}{dx} and solve:
dydx=2x(x+y)x2x2+2y(x+y)2 \frac{dy}{dx} = \frac{2x(x+y) - x^2}{x^2 + 2y(x+y)^2}
The solutions for dydx\frac{dy}{dx} are:
1. For Problem 1(a): dydx=6xy\frac{dy}{dx} = \frac{6x}{y}
2. For Problem 1(b): y=±6x6x24y' = \pm \frac{6x}{\sqrt{6x^2 - 4}}
3. For Problem 2: dydx=2x(x+y)x2x2+2y(x+y)2\frac{dy}{dx} = \frac{2x(x+y) - x^2}{x^2 + 2y(x+y)^2}

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