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

Differentiate Equation: First, we need to find the derivative of both sides of the equation with respect to x to find (dy)/(dx). The equation is -5 - x^3 - y^3 = -3x^2. Differentiate both sides with respect to x, remembering to use the chain rule for the y^3 term since y is a function of x. d/dx(-5 - x^3 - y^3) = d/dx(-3x^2) This gives us: -3x^2 - 3y^2(dy/dx) = -6xSolve for dy/dx: Now, we need to solve for (dy/dx). -3y^2(dy/dx) = -6x + 3x^2 Divide both sides by -3y^2 to isolate (dy/dx). (dy/dx) = (-6x + 3x^2) / (-3y^2)Substitute Point: Next, we substitute the point (2, -1) into the equation to find the value of (dy/dx) at that point. (dy/dx)|_((2,-1)) = (-6(2) + 3(2)^2) / (-3(-1)^2) This simplifies to: (dy/dx)|_((2,-1)) = (-12 + 12) / (-3)Calculate Value: Finally, we calculate the value. (dy/dx)|_((2,-1)) = 0 / (-3) (dy/dx)|_((2,-1)) = 0

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

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

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Math Problems

Algebra 2

Transformations of absolute value functions

Full solution

Q. If

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

then find

\frac{d y}{d x}

at the point

(2,-1)

\newline

Answer:

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

Differentiate Equation: First, we need to find the derivative of both sides of the equation with respect to $x$ to find $\frac{dy}{dx}$ . The equation is $-5 - x^3 - y^3 = -3x^2$ . Differentiate both sides with respect to $x$ , remembering to use the chain rule for the $y^3$ term since $y$ is a function of $x$ . $\frac{d}{dx}(-5 - x^3 - y^3) = \frac{d}{dx}(-3x^2)$ This gives us: $-3x^2 - 3y^2\frac{dy}{dx} = -6x$
Solve for $\frac{dy}{dx}$ : Now, we need to solve for $\frac{dy}{dx}$ .
$-3$ y^ $2$ \left(\frac{dy}{dx}\right) = $-6$ x + $3$ x^ $2$
Divide both sides by $-3$ y^ $2$ to isolate $\frac{dy}{dx}$ .
$\frac{dy}{dx}$ = \frac{ $-6$ x + $3$ x^ $2$ }{ $-3$ y^ $2$ }
Substitute Point: Next, we substitute the point $(2, -1)$ into the equation to find the value of $(dy/dx)$ at that point. $\newline$ $(dy/dx)|_{(2,-1)} = \frac{-6(2) + 3(2)^2}{-3(-1)^2}$ $\newline$ This simplifies to: $\newline$ $(dy/dx)|_{(2,-1)} = \frac{-12 + 12}{-3}$
Calculate Value: Finally, we calculate the value. $\newline$ $(\frac{dy}{dx})|_{(2,-1)} = \frac{0}{-3}$ $\newline$ $(\frac{dy}{dx})|_{(2,-1)} = 0$