Get the answer to "implicit differentiation1.if y^2=× 2.if ×^2+3×y+2y^2=3,find dy/dx at point..."

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2022-02-09

implicit differentiation1.if y^2=×

2.if ×^2+3×y+2y^2=3,find dy/dx at point (-1,2)

Vasquez

Expert2022-03-13Added 669 answers

$x^{2} + 3 x y + 2 y^{2} = 3$

Differentiate both sides of the equation.

$\frac{d}{d x} (x^{2} + 3 x y + 2 y^{2}) = \frac{d}{d x} 3$

Differentiate the left side of the equation.

$3 x y^{'} + 4 y y^{'} + 2 x + 3 y$

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

Reform the equation by setting the left side equal to the right side.

$3 x y^{'} + 4 y y^{'} + 2 x + 3 y = 0$

Solve for $y^{'}$

$y^{'} = - \frac{2 x + 3 y}{3 x + 4 y}$

Replace $y^{'}$ with $\frac{d y}{d x}$

$\frac{d y}{d x} = - \frac{2 x + 3 y}{3 x + 4 y}$

Put $(- 1, 2)$ , where $x = - 1, y = 2$

$\frac{d y}{d x} = - \frac{2 (- 1) + 3 (2)}{3 (- 1) + 4 (2)}$

$\frac{d y}{d x} = - \frac{4}{5}$

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