2022-02-09

implicit differentiation1.if y^2=×

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

Vasquez

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

Differentiate both sides of the equation.

$\frac{d}{dx}\left({x}^{2}+3xy+2{y}^{2}\right)=\frac{d}{dx}3$

Differentiate the left side of the equation.

$3x{y}^{\prime }+4y{y}^{\prime }+2x+3y$

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.

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

Solve for ${y}^{\prime }$

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

Replace ${y}^{\prime }$ with $\frac{dy}{dx}$

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

Put $\left(-1,2\right)$, where

$\frac{dy}{dx}=-\frac{2\left(-1\right)+3\left(2\right)}{3\left(-1\right)+4\left(2\right)}$

$\frac{dy}{dx}=-\frac{4}{5}$

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