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

Differentiate Equation: Differentiate both sides of the equation with respect to x using implicit differentiation. (d)/(dx)(x^2 - 3xy + y^2) = (d)/(dx)(1)Apply Product Rule: Apply the product rule to -3xy: (d)/(dx)(-3xy) = -3(dy/dx)x - 3y. So, (d)/(dx)(x^2) - 3(dy/dx)x - 3y + (d)/(dx)(y^2) = 0.Substitute Values: (d)/(dx)(x^2) = 2x, and (d)/(dx)(y^2) = 2y(dy/dx) because y is a function of x. So, 2x - 3(dy/dx)x - 3y + 2y(dy/dx) = 0.Simplify Equation: Now plug in the point (1,0) into the differentiated equation. 2(1) - 3(dy/dx)(1) - 3(0) + 2(0)(dy/dx) = 0.Solve for dy/dx: Simplify the equation: 2 - 3(dy/dx) = 0.Solve for (dy/dx): 3(dy/dx) = 2. (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)
(2)/(3)
(C) 1
(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) $\frac{2}{3}$ $\newline$ (C) $1$ $\newline$ (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)

\frac{2}{3}

\newline

(C)

1

\newline

(D)

-1

Differentiate Equation: Differentiate both sides of the equation with respect to $x$ using implicit differentiation. $\frac{d}{dx}(x^2 - 3xy + y^2) = \frac{d}{dx}(1)$
Apply Product Rule: Apply the product rule to $-3xy$ : $\frac{d}{dx}(-3xy) = -3\frac{dy}{dx}x - 3y$ . So, $\frac{d}{dx}(x^2) - 3\frac{dy}{dx}x - 3y + \frac{d}{dx}(y^2) = 0$ .
Substitute Values: $(\frac{d}{dx})(x^2) = 2x$ , and $(\frac{d}{dx})(y^2) = 2y(\frac{dy}{dx})$ because $y$ is a function of $x$ . So, $2x - 3(\frac{dy}{dx})x - 3y + 2y(\frac{dy}{dx}) = 0$ .
Simplify Equation: Now plug in the point $(1,0)$ into the differentiated equation. $\newline$ $2(1) - 3\left(\frac{dy}{dx}\right)(1) - 3(0) + 2(0)\left(\frac{dy}{dx}\right) = 0$ .
Solve for $\frac{dy}{dx}$ : Simplify the equation: $2 - 3\left(\frac{dy}{dx}\right) = 0$ .
Solve for $\frac{dy}{dx}$ : Simplify the equation: $2 - 3\left(\frac{dy}{dx}\right) = 0$ . Solve for $\frac{dy}{dx}$ : $3\left(\frac{dy}{dx}\right) = 2$ . $\frac{dy}{dx} = \frac{2}{3}$ .