for the following function
\f(x,y) = 5x^2 + 7y^2 - 3x + 4y + 2xy\
a) find the gradient

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{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Partial Derivatives" ], "new_concepts": [ "Gradient Vector" ], "current_concepts": [ "Partial Derivatives", "Gradient Vector" ] } ## Compute the partial derivative with respect to x $$ \begin{aligned} f_x(x,y) &= \frac{\partial}{\partial x}(5x^2 + 7y^2 - 3x + 4y + 2xy) \ &= 10x - 3 + 2y \end{aligned} $$ ## Compute the partial derivative with respect to y $$ \begin{aligned} f_y(x,y) &= \frac{\partial}{\partial y}(5x^2 + 7y^2 - 3x + 4y + 2xy) \ &= 14y + 4 + 2x \end{aligned} $$ ## Construct the gradient vector $$ \begin{aligned} abla f(x,y) &= \langle f_x(x,y), f_y(x,y) angle \ &= \langle 10x + 2y - 3, 2x + 14y + 4 angle \end{aligned} $$ $$ abla f(x,y) = \langle 10x + 2y - 3, 2x + 14y + 4 angle$$ { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Calculus", "Gradient Vector" ] }

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Explanation:

Compute the partial derivative with respect to x

Compute the partial derivative with respect to y

Construct the gradient vector

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