If -5x^(2)-y^(3)+4y^(2)=3 then find (dy)/(dx) at the point (1,2). Answer: (dy)/(dx)|_((1,2))=

Implicit Differentiation: To find the derivative of y with respect to x, (dy)/(dx), we need to implicitly differentiate the given equation with respect to x.Differentiate Terms: Differentiate each term of the equation with respect to x, remembering to use the chain rule for terms involving y, since y is a function of x. d/dx(-5x^2) = -10x d/dx(-y^3) = -3y^2 * (dy/dx) d/dx(4y^2) = 8y * (dy/dx) d/dx(3) = 0Combine Derivatives: Combine the derivatives to form the differentiated equation: -10x - 3y^2 * (dy/dx) + 8y * (dy/dx) = 0Isolate (dy/dx): Now, solve for (dy/dx) by isolating terms that contain (dy/dx) on one side of the equation: -3y^2 * (dy/dx) + 8y * (dy/dx) = 10xFactor Out (dy/dx): Factor out (dy/dx) from the terms on the left side of the equation: (dy/dx) * (-3y^2 + 8y) = 10xSolve for (dy/dx): Divide both sides of the equation by (-3y^2 + 8y) to solve for (dy/dx): (dy/dx) = 10x / (-3y^2 + 8y)Substitute Values: Substitute x = 1 and y = 2 into the equation to find the value of (dy/dx) at the point (1,2): (dy/dx)|_((1,2)) = 10(1) / (-3(2)^2 + 8(2))Calculate (dy/dx): Calculate the value of (dy/dx) at the point (1,2): (dy/dx)|_((1,2)) = 10 / (-3(4) + 16) (dy/dx)|_((1,2)) = 10 / (-12 + 16) (dy/dx)|_((1,2)) = 10 / 4 (dy/dx)|_((1,2)) = 2.5

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If
-5x^(2)-y^(3)+4y^(2)=3 then find
(dy)/(dx) at the point
(1,2).
Answer:
(dy)/(dx)|_((1,2))=

If $-5 x^{2}-y^{3}+4 y^{2}=3$ then find $\frac{d y}{d x}$ at the point $(1,2)$ . $\newline$ Answer: $\left.\frac{d y}{d x}\right|_{(1,2)}=$

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Algebra 1

Transformations of quadratic functions

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Q. If

-5 x^{2}-y^{3}+4 y^{2}=3

then find

\frac{d y}{d x}

at the point

(1,2)

\newline

Answer:

\left.\frac{d y}{d x}\right|_{(1,2)}=

Implicit Differentiation: To find the derivative of $y$ with respect to $x$ , $\frac{dy}{dx}$ , we need to implicitly differentiate the given equation with respect to $x$ .
Differentiate Terms: Differentiate each term of the equation with respect to $x$ , remembering to use the chain rule for terms involving $y$ , since $y$ is a function of $x$ .
$\frac{d}{dx}(-5x^2) = -10x$
$\frac{d}{dx}(-y^3) = -3y^2 \cdot \frac{dy}{dx}$
$\frac{d}{dx}(4y^2) = 8y \cdot \frac{dy}{dx}$
$\frac{d}{dx}(3) = 0$
Combine Derivatives: Combine the derivatives to form the differentiated equation: $\newline$ $-10x - 3y^2 \cdot \frac{dy}{dx} + 8y \cdot \frac{dy}{dx} = 0$
Isolate $\frac{dy}{dx}$ : Now, solve for $\frac{dy}{dx}$ by isolating terms that contain $\frac{dy}{dx}$ on one side of the equation: $\newline$ \(-3y^ $2$ \cdot \frac{dy}{dx} + $8$ y \cdot \frac{dy}{dx} = $10$ x
Factor Out $(\frac{dy}{dx}):</b> Factor out \$(\frac{dy}{dx})$ from the terms on the left side of the equation: $\newline$ (\frac{dy}{dx}) \cdot (\(-3y^ $2$ + $8$ y) = $10$ x
Solve for $(\frac{dy}{dx}):$ Divide both sides of the equation by $(-3y^2 + 8y)$ to solve for $(\frac{dy}{dx}):$ $\frac{dy}{dx} = \frac{10x}{-3y^2 + 8y}$
Substitute Values: Substitute $x = 1$ and $y = 2$ into the equation to find the value of $(\frac{dy}{dx})$ at the point $(1,2)$ : $(\frac{dy}{dx})|_{(1,2)} = \frac{10(1)}{(-3(2)^2 + 8(2))}$
Calculate $\frac{dy}{dx}$ : Calculate the value of $\frac{dy}{dx}$ at the point $(1,2)$ :
$\left.\frac{dy}{dx}\right|_{(1,2)} = \frac{10}{-3(4) + 16}$
$\left.\frac{dy}{dx}\right|_{(1,2)} = \frac{10}{-12 + 16}$
$\left.\frac{dy}{dx}\right|_{(1,2)} = \frac{10}{4}$
$\left.\frac{dy}{dx}\right|_{(1,2)} = 2.5$

If $-5 x^{2}-y^{3}+4 y^{2}=3$ then find $\frac{d y}{d x}$ at the point $(1,2)$ . $\newline$ Answer: $\left.\frac{d y}{d x}\right|_{(1,2)}=$

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If $-5 x^{2}-y^{3}+4 y^{2}=3$ then find $\frac{d y}{d x}$ at the point $(1,2)$ . $\newline$ Answer: $\left.\frac{d y}{d x}\right|_{(1,2)}=$