If y^(3)-5x^(2)+x^(3)=-5y^(2) then find (dy)/(dx) in terms of x and y. Answer: (dy)/(dx)=

Differentiate y^(3): To find the derivative (dy)/(dx), we need to differentiate both sides of the equation with respect to x, treating y as a function of x (implicit differentiation). The equation is y^(3) - 5x^(2) + x^(3) = -5y^(2). Differentiate each term with respect to x.Differentiate -5x^(2): Differentiate y^(3) with respect to x. Using the chain rule, the derivative of y^(3) with respect to x is 3y^(2)(dy/dx).Differentiate x^(3): Differentiate -5x^(2) with respect to x. The derivative of -5x^(2) with respect to x is -10x.Differentiate -5y^(2): Differentiate x^(3) with respect to x. The derivative of x^(3) with respect to x is 3x^(2).Combine differentiated terms: Differentiate -5y^(2) with respect to x. Using the chain rule, the derivative of -5y^(2) with respect to x is -10y(dy/dx).Isolate (dy)/(dx): Combine all the differentiated terms to form the derivative of the entire equation. 3y^(2)(dy/dx) - 10x + 3x^(2) = -10y(dy/dx).Factor out (dy)/(dx): Now, we need to solve for (dy)/(dx). Rearrange the terms to isolate (dy)/(dx) on one side of the equation. 3y^(2)(dy/dx) + 10y(dy/dx) = 10x - 3x^(2).Divide to solve for (dy)/(dx): Factor out (dy)/(dx) from the left side of the equation. (dy/dx)(3y^(2) + 10y) = 10x - 3x^(2).Divide both sides of the equation by (3y^(2) + 10y) to solve for (dy)/(dx). (dy/dx) = (10x - 3x^(2)) / (3y^(2) + 10y).

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If
y^(3)-5x^(2)+x^(3)=-5y^(2) then find
(dy)/(dx) in terms of
x and
y.
Answer:
(dy)/(dx)=

If $y^{3}-5 x^{2}+x^{3}=-5 y^{2}$ then find $\frac{d y}{d x}$ in terms of $x$ and $y$ . $\newline$ Answer: $\frac{d y}{d x}=$

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Algebra 1

Transformations of quadratic functions

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Q. If

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

then find

\frac{d y}{d x}

in terms of

x

and

y

\newline

Answer:

\frac{d y}{d x}=

Differentiate $y^3$ : To find the derivative $\frac{dy}{dx}$ , we need to differentiate both sides of the equation with respect to $x$ , treating $y$ as a function of $x$ (implicit differentiation). $\newline$ The equation is $y^3 - 5x^2 + x^3 = -5y^2$ . $\newline$ Differentiate each term with respect to $x$ .
Differentiate $-5x^{2}$ : Differentiate $y^{3}$ with respect to $x$ . Using the chain rule, the derivative of $y^{3}$ with respect to $x$ is $3y^{2}(\frac{dy}{dx})$ .
Differentiate $x^{3}$ : Differentiate $-5x^{2}$ with respect to $x$ . The derivative of $-5x^{2}$ with respect to $x$ is $-10x$ .
Differentiate $-5y^{2}$ : Differentiate $x^{3}$ with respect to $x$ . The derivative of $x^{3}$ with respect to $x$ is $3x^{2}$ .
Combine differentiated terms: Differentiate $-5y^{2}$ with respect to $x$ . Using the chain rule, the derivative of $-5y^{2}$ with respect to $x$ is $-10y\frac{dy}{dx}$ .
Isolate (\frac{dy}{dx}): Combine all the differentiated terms to form the derivative of the entire equation.$\newline\$3y^{2}(\frac{dy}{dx}) - 10x + 3x^{2} = -10y(\frac{dy}{dx}).$
Factor out $(dy)/(dx)$: Now, we need to solve for $(dy)/(dx)$. Rearrange the terms to isolate $(dy)/(dx)$ on one side of the equation. $3y^{2}(dy/dx) + 10y(dy/dx) = 10x - 3x^{2}$.
Divide to solve for $(\frac{dy}{dx}):$ Factor out $(\frac{dy}{dx})$ from the left side of the equation.$\newline$$(\frac{dy}{dx})(3y^{2} + 10y) = 10x - 3x^{2}.$
Divide to solve for $(dy)/(dx)$: Factor out $(dy)/(dx)$ from the left side of the equation.\[(dy/dx)(3y^{2} + 10y) = 10x - 3x^{2}\].Divide both sides of the equation by $(3y^{2} + 10y)$ to solve for $(dy)/(dx)$.\[(dy/dx) = \frac{10x - 3x^{2}}{3y^{2} + 10y}\].