x^(2)-3xy+y^(2)=1 Find the value of (dy)/(dx) at the point (1,0). Choose 1 answer: (A) (2)/(3) (B) 1 (C) -(2)/(3) (D) -1

Differentiate Equation Implicitly: Implicitly differentiate both sides of the equation with respect to x. (d)/(dx)(x^(2)-3xy+y^(2)) = (d)/(dx)(1)Apply Product Rule for -3xy: Use the product rule for -3xy: (d)/(dx)(-3xy) = -3(dy/dx)x - 3y. So, (d)/(dx)(x^(2)) - 3(dy/dx)x - 3y + (d)/(dx)(y^(2)) = 0Differentiate x^(2) and y^(2): Differentiate x^(2) to get 2x and y^(2) to get 2y(dy/dx). 2x - 3(dy/dx)x - 3y + 2y(dy/dx) = 0Solve for dy/dx: Rearrange the terms to solve for dy/dx. 2x - 3y = 3(dy/dx)x - 2y(dy/dx)Factor out dy/dx: Factor out dy/dx on the right side. 2x - 3y = dy/dx(3x - 2y)Isolate dy/dx: Divide both sides by (3x - 2y) to isolate dy/dx. (dy/dx) = (2x - 3y) / (3x - 2y)Plug in Point: Plug in the point (1,0) into the equation. (dy/dx) = (2(1) - 3(0)) / (3(1) - 2(0))Simplify Equation: Simplify the equation. (dy/dx) = (2 - 0) / (3 - 0)Calculate Final Value: Calculate the final value. (dy/dx) = 2 / 3

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x^(2)-3xy+y^(2)=1
Find the value of
(dy)/(dx) at the point
(1,0).
Choose 1 answer:
(A)
(2)/(3)
(B) 1
(C)
-(2)/(3)
(D) -1

$x^{2}-3 x y+y^{2}=1$ $\newline$ Find the value of $\frac{d y}{d x}$ at the point $(1,0)$ . $\newline$ Choose $1$ answer: $\newline$ (A) $\frac{2}{3}$ $\newline$ (B) $1$ $\newline$ (C) $-\frac{2}{3}$ $\newline$ (D) $-1$

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Calculus

Find derivatives of using multiple formulae

Full solution

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

\newline

Find the value of

\frac{d y}{d x}

at the point

(1,0)

\newline

Choose

1

answer:

\newline

(A)

\frac{2}{3}

\newline

(B)

1

\newline

(C)

-\frac{2}{3}

\newline

(D)

-1

Differentiate Equation Implicitly: Implicitly differentiate both sides of the equation with respect to $x$ . $\frac{d}{dx}(x^{2}-3xy+y^{2}) = \frac{d}{dx}(1)$
Apply Product Rule for $-3xy$ : Use the product rule for $-3xy$ : $\frac{d}{dx}(-3xy) = -3\frac{dy}{dx}x - 3y$ . So, $\frac{d}{dx}(x^{2}) - 3\frac{dy}{dx}x - 3y + \frac{d}{dx}(y^{2}) = 0$
Differentiate $x^{2}$ and $y^{2}$ : Differentiate $x^{2}$ to get $2x$ and $y^{2}$ to get $2y\frac{dy}{dx}$ . $\newline$ $2x - 3\frac{dy}{dx}x - 3y + 2y\frac{dy}{dx} = 0$
Solve for $\frac{dy}{dx}$ : Rearrange the terms to solve for $\frac{dy}{dx}$ . $2x - 3y = 3\left(\frac{dy}{dx}\right)x - 2y\left(\frac{dy}{dx}\right)$
Factor out dy/dx: Factor out $\frac{dy}{dx}$ on the right side. $\newline$ $2x - 3y = \frac{dy}{dx}(3x - 2y)$
Isolate $\frac{dy}{dx}$ : Divide both sides by $(3x - 2y)$ to isolate $\frac{dy}{dx}$ . $\frac{dy}{dx} = \frac{2x - 3y}{3x - 2y}$
Plug in Point: Plug in the point $(1,0)$ into the equation. $\newline$ $(\frac{dy}{dx}) = \frac{(2(1) - 3(0))}{(3(1) - 2(0))}$
Simplify Equation: Simplify the equation. $\frac{dy}{dx} = \frac{2 - 0}{3 - 0}$
Calculate Final Value: Calculate the final value. $\frac{dy}{dx} = \frac{2}{3}$