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

Implicit Differentiation: To find the derivative of y with respect to x, (dy)/(dx), we need to implicitly differentiate both sides of the equation with respect to x. Given equation: x^(3) + 4 - 5x^(2) = -5y + y^(2) Differentiate both sides with respect to x: d/dx [x^(3) + 4 - 5x^(2)] = d/dx [-5y + y^(2)] Using the power rule and chain rule, we get: 3x^(2) - 10x = -5(dy/dx) + 2y(dy/dx)Isolating (dy/dx): Now we need to solve for (dy)/(dx) by isolating it on one side of the equation. 3x^(2) - 10x = -5(dy/dx) + 2y(dy/dx) Combine like terms: (dy/dx)(2y - 5) = 3x^(2) - 10x Now, divide both sides by (2y - 5) to solve for (dy)/(dx): (dy/dx) = (3x^(2) - 10x) / (2y - 5)Evaluate at (1,5): We need to evaluate (dy)/(dx) at the point (1,5). Substitute x = 1 and y = 5 into the equation: (dy/dx) = (3(1)^(2) - 10(1)) / (2(5) - 5) Simplify the equation: (dy/dx) = (3 - 10) / (10 - 5) (dy/dx) = (-7) / 5

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

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

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Algebra 2

Transformations of absolute value functions

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Q. If

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

then find

\frac{d y}{d x}

at the point

(1,5)

\newline

Answer:

\left.\frac{d y}{d x}\right|_{(1,5)}=

Implicit Differentiation: To find the derivative of $y$ with respect to $x$ , $\frac{dy}{dx}$ , we need to implicitly differentiate both sides of the equation with respect to $x$ . $\newline$ Given equation: $x^{3} + 4 - 5x^{2} = -5y + y^{2}$ $\newline$ Differentiate both sides with respect to $x$ : $\newline$ $\frac{d}{dx} [x^{3} + 4 - 5x^{2}] = \frac{d}{dx} [-5y + y^{2}]$ $\newline$ Using the power rule and chain rule, we get: $\newline$ $3x^{2} - 10x = -5\frac{dy}{dx} + 2y\frac{dy}{dx}$
Isolating $\frac{dy}{dx}$ : Now we need to solve for $\frac{dy}{dx}$ by isolating it on one side of the equation. $\newline$ $3x^{2} - 10x = -5\frac{dy}{dx} + 2y\frac{dy}{dx}$ $\newline$ Combine like terms: $\newline$ $\frac{dy}{dx}(2y - 5) = 3x^{2} - 10x$ $\newline$ Now, divide both sides by $(2y - 5)$ to solve for $\frac{dy}{dx}$ : $\newline$ $\frac{dy}{dx} = \frac{3x^{2} - 10x}{2y - 5}$
Evaluate at $(1,5)$ : We need to evaluate $\frac{dy}{dx}$ at the point $(1,5)$ . $\newline$ Substitute $x = 1$ and $y = 5$ into the equation: $\newline$ $\frac{dy}{dx} = \frac{3(1)^{2} - 10(1)}{2(5) - 5}$ $\newline$ Simplify the equation: $\newline$ $\frac{dy}{dx} = \frac{3 - 10}{10 - 5}$ $\newline$ $\frac{dy}{dx} = \frac{-7}{5}$