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Let's compute the first-order partial derivatives of the given function [tex]\( f(x, y) = 2x(y - 3x) - 4y \)[/tex]. A partial derivative of a function with respect to a variable treats all other variables as constants.
### Step 1: Compute [tex]\(\frac{\partial f}{\partial x}\)[/tex]
To find the partial derivative with respect to [tex]\(x\)[/tex], we need to differentiate [tex]\( f(x, y) \)[/tex] holding [tex]\( y \)[/tex] constant.
The original function is:
[tex]\[ f(x, y) = 2x(y - 3x) - 4y \][/tex]
First, expand the function:
[tex]\[ f(x, y) = 2x \cdot y - 2x \cdot 3x - 4y \][/tex]
[tex]\[ f(x, y) = 2xy - 6x^2 - 4y \][/tex]
Now, differentiate with respect to [tex]\( x \)[/tex]:
[tex]\[ \frac{\partial f}{\partial x} = \frac{\partial}{\partial x} (2xy - 6x^2 - 4y) \][/tex]
The derivative of each term is:
[tex]\[ \frac{\partial}{\partial x} (2xy) = 2y \][/tex]
[tex]\[ \frac{\partial}{\partial x} (-6x^2) = -12x \][/tex]
[tex]\[ \frac{\partial}{\partial x} (-4y) = 0 \][/tex]
So, combining these results:
[tex]\[ \frac{\partial f}{\partial x} = 2y - 12x \][/tex]
### Step 2: Compute [tex]\(\frac{\partial f}{\partial y}\)[/tex]
To find the partial derivative with respect to [tex]\( y \)[/tex], we need to differentiate [tex]\( f(x, y) \)[/tex] holding [tex]\( x \)[/tex] constant.
Using the expanded form:
[tex]\[ f(x, y) = 2xy - 6x^2 - 4y \][/tex]
Differentiate with respect to [tex]\( y \)[/tex]:
[tex]\[ \frac{\partial f}{\partial y} = \frac{\partial}{\partial y} (2xy - 6x^2 - 4y) \][/tex]
The derivative of each term is:
[tex]\[ \frac{\partial}{\partial y} (2xy) = 2x \][/tex]
[tex]\[ \frac{\partial}{\partial y} (-6x^2) = 0 \][/tex]
[tex]\[ \frac{\partial}{\partial y} (-4y) = -4 \][/tex]
So, combining these results:
[tex]\[ \frac{\partial f}{\partial y} = 2x - 4 \][/tex]
### Summary
Thus, the first-order partial derivatives of the function [tex]\( f(x, y) = 2x(y - 3x) - 4y \)[/tex] are:
[tex]\[ \frac{\partial f}{\partial x} = -12x + 2y \][/tex]
[tex]\[ \frac{\partial f}{\partial y} = 2x - 4 \][/tex]
These results simplify to:
[tex]\[ \frac{\partial f}{\partial x} = -12x + 2y \][/tex]
[tex]\[ \frac{\partial f}{\partial y} = 2x - 4 \][/tex]
### Step 1: Compute [tex]\(\frac{\partial f}{\partial x}\)[/tex]
To find the partial derivative with respect to [tex]\(x\)[/tex], we need to differentiate [tex]\( f(x, y) \)[/tex] holding [tex]\( y \)[/tex] constant.
The original function is:
[tex]\[ f(x, y) = 2x(y - 3x) - 4y \][/tex]
First, expand the function:
[tex]\[ f(x, y) = 2x \cdot y - 2x \cdot 3x - 4y \][/tex]
[tex]\[ f(x, y) = 2xy - 6x^2 - 4y \][/tex]
Now, differentiate with respect to [tex]\( x \)[/tex]:
[tex]\[ \frac{\partial f}{\partial x} = \frac{\partial}{\partial x} (2xy - 6x^2 - 4y) \][/tex]
The derivative of each term is:
[tex]\[ \frac{\partial}{\partial x} (2xy) = 2y \][/tex]
[tex]\[ \frac{\partial}{\partial x} (-6x^2) = -12x \][/tex]
[tex]\[ \frac{\partial}{\partial x} (-4y) = 0 \][/tex]
So, combining these results:
[tex]\[ \frac{\partial f}{\partial x} = 2y - 12x \][/tex]
### Step 2: Compute [tex]\(\frac{\partial f}{\partial y}\)[/tex]
To find the partial derivative with respect to [tex]\( y \)[/tex], we need to differentiate [tex]\( f(x, y) \)[/tex] holding [tex]\( x \)[/tex] constant.
Using the expanded form:
[tex]\[ f(x, y) = 2xy - 6x^2 - 4y \][/tex]
Differentiate with respect to [tex]\( y \)[/tex]:
[tex]\[ \frac{\partial f}{\partial y} = \frac{\partial}{\partial y} (2xy - 6x^2 - 4y) \][/tex]
The derivative of each term is:
[tex]\[ \frac{\partial}{\partial y} (2xy) = 2x \][/tex]
[tex]\[ \frac{\partial}{\partial y} (-6x^2) = 0 \][/tex]
[tex]\[ \frac{\partial}{\partial y} (-4y) = -4 \][/tex]
So, combining these results:
[tex]\[ \frac{\partial f}{\partial y} = 2x - 4 \][/tex]
### Summary
Thus, the first-order partial derivatives of the function [tex]\( f(x, y) = 2x(y - 3x) - 4y \)[/tex] are:
[tex]\[ \frac{\partial f}{\partial x} = -12x + 2y \][/tex]
[tex]\[ \frac{\partial f}{\partial y} = 2x - 4 \][/tex]
These results simplify to:
[tex]\[ \frac{\partial f}{\partial x} = -12x + 2y \][/tex]
[tex]\[ \frac{\partial f}{\partial y} = 2x - 4 \][/tex]
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