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

Apply Implicit Differentiation: First, we need to differentiate both sides of the equation with respect to x, using implicit differentiation, because y is a function of x. Differentiate -x^2 to get -2x. Differentiate -5y^2 with respect to x to get -10y(dy/dx). Differentiate y^3 with respect to x to get 3y^2(dy/dx). The differentiated equation is: -2x = -10y(dy/dx) + 3y^2(dy/dx).Factor Out (dy/dx): Now, we need to solve for (dy/dx) by factoring it out from the terms on the right side of the equation. Group the terms with (dy/dx) together: -2x = (dy/dx)(-10y + 3y^2).Isolate (dy/dx): Next, isolate (dy/dx) by dividing both sides of the equation by (-10y + 3y^2). (dy/dx) = -2x / (-10y + 3y^2).Substitute Point: Now we need to substitute the point (2,1) into the equation to find the value of (dy/dx) at that point. Substitute x = 2 and y = 1 into the equation: (dy/dx) = -2(2) / (-10(1) + 3(1)^2).Calculate (dy/dx): Perform the calculations to find the value of (dy/dx). (dy/dx) = -4 / (-10 + 3) = -4 / (-7) = 4/7.

Home

Math Problems

Grade 8

Evaluate a nonlinear function

Full solution

Q. If

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

then find

\frac{d y}{d x}

at the point

(2,1)

\newline

Answer:

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

Apply Implicit Differentiation: First, we need to differentiate both sides of the equation with respect to $x$ , using implicit differentiation, because $y$ is a function of $x$ . Differentiate $-x^2$ to get $-2x$ . Differentiate $-5y^2$ with respect to $x$ to get $-10y\frac{dy}{dx}$ . Differentiate $y^3$ with respect to $x$ to get $y$ $0$ . The differentiated equation is: $y$ $1$ .
Factor Out $(\frac{dy}{dx}):</b> Now, we need to solve for \$(\frac{dy}{dx})$ by factoring it out from the terms on the right side of the equation. $\newline$ Group the terms with $(\frac{dy}{dx})$ together: $-2x = (\frac{dy}{dx})(-10y + 3y^2)$ .
Isolate $(\frac{dy}{dx}):</b> Next, isolate \$(\frac{dy}{dx})$ by dividing both sides of the equation by $(-10y + 3y^2)$ . $\newline$ $(\frac{dy}{dx}) = \frac{-2x}{(-10y + 3y^2)}$ .
Substitute Point: Now we need to substitute the point $(2,1)$ into the equation to find the value of $(\frac{dy}{dx})$ at that point. $\newline$ Substitute $x = 2$ and $y = 1$ into the equation: $(\frac{dy}{dx}) = \frac{-2(2)}{(-10(1) + 3(1)^2)}$ .
Calculate (\frac{dy}{dx}): Perform the calculations to find the value of \$(\frac{dy}{dx}).$\newline\$(\frac{dy}{dx}) = \frac{-4}{(-10 + 3)} = \frac{-4}{(-7)} = \frac{4}{7}.$

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

Full solution

More problems from Evaluate a nonlinear function

Related topics

If −x2=−5y2+y3 -x^{2}=-5 y^{2}+y^{3} −x2=−5y2+y3 then find dydx \frac{d y}{d x} dxdy​ at the point (2,1) (2,1) (2,1).\newlineAnswer: dydx∣(2,1)= \left.\frac{d y}{d x}\right|_{(2,1)}= dxdy​∣∣​(2,1)​=

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