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Given the function f(x,y)=x^{2}+3\ln{(x)}-xy. A) Find f(1,0) B) Find f(2,2)

Question:

Given the function {eq}f(x,y)=x^{2}+3\ln{(x)}-xy {/eq}.

A) Find f(1,0)

B) Find f(2,2)

Function value:

If {eq}z = f(x, y) {/eq} is a function of two independent variables x and y, then

the function value at point {eq}(a, b) {/eq} is:

{eq}z = f(a, b) {/eq}

The partial differentiation of {eq}f(x, y) {/eq}:

The partial differentiation of z with respect to x when y is constant

{eq}\displaystyle \frac{\partial z}{\partial x} = \frac{\partial}{\partial x}[f(x, y)] {/eq}

The partial differentiation of z with respect to y when x is constant

{eq}\displaystyle \frac{\partial z}{\partial y} = \frac{\partial}{\partial y}[f(x, y)] {/eq}

For example: {eq}z = xy {/eq}

Take the partial differentiation of z with respect to x when y is constant

{eq}\displaystyle \frac{\partial z}{\partial x} = \frac{\partial}{\partial x}[xy] = y \frac{\partial}{\partial x}[x] = y {/eq}

Take the partial differentiation of z with respect to y when x is constant

{eq}\displaystyle \frac{\partial z}{\partial y} = \frac{\partial}{\partial y}[xy] = x \frac{\partial}{\partial y}[y] = x {/eq}

Answer and Explanation:

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Given:

{eq}f(x, y) = x^{2} + 3 \ln (x) - xy {/eq}

A. We will computer the function at point {eq}(1, 0) {/eq}

Plug the point in the given...

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Partial Derivative: Definition, Rules & Examples

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Chapter 18 / Lesson 12
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When a function depends on more than one variable, we can use the partial derivative to determine how that function changes with respect to one variable at a time. In this lesson, we use examples to define partial derivatives and to explain the rules for evaluating them.


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