If -x^(2)-x^(3)y+1=-y then find (dy)/(dx) in terms of x and y. Answer: (dy)/(dx)=

Write Equation Differentiation: Write down the given equation and differentiate both sides with respect to x. Given equation: -x^2 - x^3y + 1 = -y Differentiate both sides with respect to x using the product rule for the term -x^3y and the chain rule for the term -y.Differentiate Left Side: Differentiate each term on the left side of the equation. Differentiate -x^2 with respect to x to get -2x. Differentiate -x^3y with respect to x using the product rule: d/dx(-x^3y) = -x^3(dy/dx) - 3x^2y. Differentiate +1 with respect to x to get 0, since it is a constant.Differentiate Right Side: Differentiate the right side of the equation. Differentiate -y with respect to x using the chain rule: d/dx(-y) = -(dy/dx).Write Differentiated Equation: Write down the differentiated equation. The differentiated equation is: -2x - x^3(dy/dx) - 3x^2y = -(dy/dx).Isolate Terms: Isolate terms containing (dy/dx) on one side of the equation. Add x^3(dy/dx) to both sides and add (dy/dx) to both sides to get: -2x - 3x^2y = x^3(dy/dx) + (dy/dx).Factor Out (dy/dx): Factor out (dy/dx) from the right side of the equation. We get: -2x - 3x^2y = (dy/dx)(x^3 + 1).Solve for (dy/dx): Solve for (dy/dx). Divide both sides by (x^3 + 1) to isolate (dy/dx): (dy/dx) = (-2x - 3x^2y) / (x^3 + 1).

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If
-x^(2)-x^(3)y+1=-y then find
(dy)/(dx) in terms of
x and
y.
Answer:
(dy)/(dx)=

If $-x^{2}-x^{3} y+1=-y$ then find $\frac{d y}{d x}$ in terms of $x$ and $y$ . $\newline$ Answer: $\frac{d y}{d x}=$

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Algebra 1

Transformations of quadratic functions

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Q. If

-x^{2}-x^{3} y+1=-y

then find

\frac{d y}{d x}

in terms of

x

and

y

\newline

Answer:

\frac{d y}{d x}=

Write Equation Differentiation: Write down the given equation and differentiate both sides with respect to $x$ . $\newline$ Given equation: $-x^2 - x^3y + 1 = -y$ $\newline$ Differentiate both sides with respect to $x$ using the product rule for the term $-x^3y$ and the chain rule for the term $-y$ .
Differentiate Left Side: Differentiate each term on the left side of the equation. $\newline$ Differentiate $-x^2$ with respect to $x$ to get $-2x$ . $\newline$ Differentiate $-x^3y$ with respect to $x$ using the product rule: $\frac{d}{dx}(-x^3y) = -x^3\frac{dy}{dx} - 3x^2y$ . $\newline$ Differentiate $+1$ with respect to $x$ to get $0$ , since it is a constant.
Differentiate Right Side: Differentiate the right side of the equation. $\newline$ Differentiate $-y$ with respect to $x$ using the chain rule: $\frac{d}{dx}(-y) = -(\frac{dy}{dx})$ .
Write Differentiated Equation: Write down the differentiated equation. $\newline$ The differentiated equation is: $-2x - x^3\frac{dy}{dx} - 3x^2y = -\frac{dy}{dx}$ .
Isolate Terms: Isolate terms containing $\frac{dy}{dx}$ on one side of the equation. $\newline$ Add $x^3\frac{dy}{dx}$ to both sides and add $\frac{dy}{dx}$ to both sides to get: $-2x - 3x^2y = x^3\frac{dy}{dx} + \frac{dy}{dx}$ .
Factor Out $(\frac{dy}{dx}):</b> Factor out \$(\frac{dy}{dx})$ from the right side of the equation. $\newline$ We get: \(-2x - $3$ x^ $2$ y = (\frac{dy}{dx})(x^ $3$ + $1$ ).
Solve for $(\frac{dy}{dx}):</b> Solve for \$(\frac{dy}{dx})$ . $\newline$ Divide both sides by $(x^3 + 1)$ to isolate $(\frac{dy}{dx})$ : $(\frac{dy}{dx}) = \frac{-2x - 3x^2y}{x^3 + 1}$ .

If $-x^{2}-x^{3} y+1=-y$ then find $\frac{d y}{d x}$ in terms of $x$ and $y$ . $\newline$ Answer: $\frac{d y}{d x}=$

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If $-x^{2}-x^{3} y+1=-y$ then find $\frac{d y}{d x}$ in terms of $x$ and $y$ . $\newline$ Answer: $\frac{d y}{d x}=$