Question

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

Explanation:

<pre_analysis>
{
"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"
]
}
</pre_analysis>

<reasoning>

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

\[

$$\begin{aligned} abla f(x,y) &= \langle f_x(x,y), f_y(x,y) angle \\ &= \langle 10x + 2y - 3, 2x + 14y + 4 angle \end{aligned}$$

\]
</reasoning>

<answer>
\[
abla f(x,y) = \langle 10x + 2y - 3, 2x + 14y + 4
angle\]
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Free Response",
"knowledge_point": [
"Mathematics",
"Calculus",
"Gradient Vector"
]
}
</post_analysis>

Answer:

<pre_analysis>
{
"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"
]
}
</pre_analysis>

<reasoning>

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

\[

$$\begin{aligned} abla f(x,y) &= \langle f_x(x,y), f_y(x,y) angle \\ &= \langle 10x + 2y - 3, 2x + 14y + 4 angle \end{aligned}$$

\]
</reasoning>

<answer>
\[
abla f(x,y) = \langle 10x + 2y - 3, 2x + 14y + 4
angle\]
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Free Response",
"knowledge_point": [
"Mathematics",
"Calculus",
"Gradient Vector"
]
}
</post_analysis>