Use implicit differentiation to find (dy)/(dx) at the given point: -2y^(2)+4=4x" at "(-1,2)

Differentiate both sides: We are given the equation -2y^2 + 4 = 4x and we need to find the derivative of y with respect to x, (dy)/(dx), at the point (-1,2). We will use implicit differentiation to find (dy)/(dx). First, differentiate both sides of the equation with respect to x. The left side of the equation is -2y^2 + 4. When we differentiate -2y^2 with respect to x, we treat y as a function of x (y(x)), so we need to use the chain rule. The derivative of -2y^2 is -4y(dy)/(dx). The derivative of the constant 4 is 0. The right side of the equation is 4x. The derivative of 4x with respect to x is 4. So, differentiating both sides gives us -4y(dy)/(dx) = 4.Solve for (dy)/(dx): Now we need to solve for (dy)/(dx). Divide both sides of the equation by -4y to isolate (dy)/(dx) on one side. (dy)/(dx) = 4 / (-4y) Simplify the right side of the equation. (dy)/(dx) = -1/ySubstitute given point: Next, we substitute the given point (-1,2) into the equation to find the value of (dy)/(dx) at that specific point. (dy)/(dx) at (-1,2) is -1/2.Find value at point: We have found the value of (dy)/(dx) at the point (-1,2) for the given equation.

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Use implicit differentiation to find $\frac{dy}{dx}$ at the given point: $\newline$ $-2y^{2}+4=4x$ at $(-1,2)$

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Q. Use implicit differentiation to find

\frac{dy}{dx}

at the given point:

\newline

-2y^{2}+4=4x

(-1,2)

Differentiate both sides: We are given the equation $-2y^2 + 4 = 4x$ and we need to find the derivative of $y$ with respect to $x$ , $\frac{dy}{dx}$ , at the point $(-1,2)$ . We will use implicit differentiation to find $\frac{dy}{dx}$ . First, differentiate both sides of the equation with respect to $x$ . The left side of the equation is $-2y^2 + 4$ . When we differentiate $-2y^2$ with respect to $x$ , we treat $y$ as a function of $x$ ( $y$ $2$ ), so we need to use the chain rule. The derivative of $-2y^2$ is $y$ $4$ . The derivative of the constant $y$ $5$ is $y$ $6$ . The right side of the equation is $y$ $7$ . The derivative of $y$ $7$ with respect to $x$ is $y$ $5$ . So, differentiating both sides gives us $x$ $1$ .
Solve for (\frac{dy}{dx}): Now we need to solve for \$(\frac{dy}{dx})\.$\newlineDivide both sides of the equation by \$-4y$ to isolate $(\frac{dy}{dx})$ on one side.$\newline$$(\frac{dy}{dx}) = \frac{4}{(-4y)}$$\newline$Simplify the right side of the equation.$\newline$$(\frac{dy}{dx}) = -\frac{1}{y}$
Substitute given point: Next, we substitute the given point $(-1,2)$ into the equation to find the value of $\frac{dy}{dx}$ at that specific point.$\newline$$\frac{dy}{dx}$ at $(-1,2)$ is $-\frac{1}{2}$.
Find value at point: We have found the value of $\frac{dy}{dx}$ at the point $(-1,2)$ for the given equation.