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

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Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Partial Derivatives" ], "new_concepts": [ "Implicit Differentiation" ], "current_concepts": [ "Partial Derivatives", "Implicit Differentiation" ] } ## 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 $$ \begin{aligned} \frac{dy}{dx} &= -\frac{F_x}{F_y} \ \frac{dy}{dx} &= -\frac{10x + 2y - 3}{2x + 14y + 4} \end{aligned} $$ $$ \frac{dy}{dx} = -\frac{10x + 2y - 3}{2x + 14y + 4} $$ { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Calculus", "Implicit Differentiation" ] }

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Explanation:

Find the partial derivatives with respect to x and y

Apply the implicit differentiation formula

Answer: