Question

for the following
\\5x^2 + 7y^2 - 3x + 4y + 2xy = 0\\
find \\(\frac{dy}{dx}\\)

Explanation:

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<reasoning>

Find the partial derivatives with respect to x and y

\[

$$\begin{aligned} F(x, y) &= 5x^2 + 7y^2 - 3x + 4y + 2xy = 0 \\ F_x &= \frac{\partial F}{\partial x} = 10x - 3 + 2y \\ F_y &= \frac{\partial F}{\partial y} = 14y + 4 + 2x \end{aligned}$$

\]

Apply the implicit differentiation formula

\[

$$\begin{aligned} \frac{dy}{dx} &= -\frac{F_x}{F_y} \\ \frac{dy}{dx} &= -\frac{10x + 2y - 3}{2x + 14y + 4} \end{aligned}$$

\]
</reasoning>

<answer>
\[
\frac{dy}{dx} = -\frac{10x + 2y - 3}{2x + 14y + 4}
\]
</answer>

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Answer:

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],
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</pre_analysis>

<reasoning>

Find the partial derivatives with respect to x and y

\[

$$\begin{aligned} F(x, y) &= 5x^2 + 7y^2 - 3x + 4y + 2xy = 0 \\ F_x &= \frac{\partial F}{\partial x} = 10x - 3 + 2y \\ F_y &= \frac{\partial F}{\partial y} = 14y + 4 + 2x \end{aligned}$$

\]

Apply the implicit differentiation formula

\[

$$\begin{aligned} \frac{dy}{dx} &= -\frac{F_x}{F_y} \\ \frac{dy}{dx} &= -\frac{10x + 2y - 3}{2x + 14y + 4} \end{aligned}$$

\]
</reasoning>

<answer>
\[
\frac{dy}{dx} = -\frac{10x + 2y - 3}{2x + 14y + 4}
\]
</answer>

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