If xy+2x=-2+2y then find (dy)/(dx) in terms of x and y. Answer: (dy)/(dx)=

Given equation differentiation: We are given the equation xy + 2x = -2 + 2y. To find (dy)/(dx), we need to differentiate both sides of the equation with respect to x, treating y as a function of x (implicit differentiation).Left side differentiation: Differentiate the left side of the equation with respect to x. The term xy is a product of x and y, so we use the product rule: d(uv)/dx = u(dv/dx) + v(du/dx). Here, u = x and v = y, so we get x(dy/dx) + y(1). The term 2x is straightforward: the derivative of 2x with respect to x is 2.Right side differentiation: Differentiate the right side of the equation with respect to x. The term -2 is a constant, so its derivative is 0. The term 2y is a function of x, so we get 2(dy/dx).Write down differentiated equation: Now we write down the differentiated equation: x(dy/dx) + y + 2 = 2(dy/dx).Solve for dy/dx: We need to solve for (dy/dx). To do this, we'll collect all the terms containing (dy/dx) on one side and the other terms on the opposite side. Subtract x(dy/dx) from both sides to get: 2(dy/dx) - x(dy/dx) = y + 2.Factor out dy/dx: Factor out (dy/dx) from the left side: (dy/dx)(2 - x) = y + 2.Divide to solve for dy/dx: Divide both sides by (2 - x) to solve for (dy/dx): (dy/dx) = (y + 2) / (2 - x).

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If
xy+2x=-2+2y then find
(dy)/(dx) in terms of
x and
y.
Answer:
(dy)/(dx)=

If $x y+2 x=-2+2 y$ then find $\frac{d y}{d x}$ in terms of $x$ and $y$ . $\newline$ Answer: $\frac{d y}{d x}=$

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Q. If

x y+2 x=-2+2 y

then find

\frac{d y}{d x}

in terms of

x

and

y

\newline

Answer:

\frac{d y}{d x}=

Given equation differentiation: We are given the equation $xy + 2x = -2 + 2y$ . To find $\frac{dy}{dx}$ , we need to differentiate both sides of the equation with respect to $x$ , treating $y$ as a function of $x$ (implicit differentiation).
Left side differentiation: Differentiate the left side of the equation with respect to $x$ . The term $xy$ is a product of $x$ and $y$ , so we use the product rule: $\frac{d(uv)}{dx} = u\frac{dv}{dx} + v\frac{du}{dx}$ . Here, $u = x$ and $v = y$ , so we get $x\frac{dy}{dx} + y(1)$ . The term $2x$ is straightforward: the derivative of $2x$ with respect to $x$ is $xy$ $1$ .
Right side differentiation: Differentiate the right side of the equation with respect to $x$ . The term $-2$ is a constant, so its derivative is $0$ . The term $2y$ is a function of $x$ , so we get $2\frac{dy}{dx}$ .
Write down differentiated equation: Now we write down the differentiated equation: $x\frac{dy}{dx} + y + 2 = 2\frac{dy}{dx}$ .
Solve for $\frac{dy}{dx}$ : We need to solve for $\frac{dy}{dx}$ . To do this, we'll collect all the terms containing $\frac{dy}{dx}$ on one side and the other terms on the opposite side. Subtract $x\frac{dy}{dx}$ from both sides to get: $2\frac{dy}{dx} - x\frac{dy}{dx} = y + 2$ .
Factor out $\frac{dy}{dx}$ : Factor out $\left(\frac{dy}{dx}\right)$ from the left side: $\left(\frac{dy}{dx}\right)(2 - x) = y + 2$ .
Divide to solve for $\frac{dy}{dx}$ : Divide both sides by $(2 - x)$ to solve for $\frac{dy}{dx}$ : $\frac{dy}{dx} = \frac{y + 2}{2 - x}$ .