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Chapter 21: Problem 3

$$ \text { If } z=3 x^{2}+7 x y-y^{2} \text { find } \frac{\partial^{2} z}{\partial y \partial x} \text { and } \frac{\partial^{2} z}{\partial x \partial y} \text {. } $$

Short Answer

Expert verified
Answer: The second-order mixed partial derivatives are equal, and both are equal to 7: $$\frac{\partial^2 z}{\partial y \partial x} = \frac{\partial^2 z}{\partial x \partial y} = 7$$

Step by step solution

01

Find the first-order partial derivatives of z with respect to x and y.

To find the first-order partial derivatives, we will treat one variable as constant and differentiate with respect to the other. First, let's find the partial derivative of z with respect to x, denoted as \(\frac{\partial z}{\partial x}\): $$ \frac{\partial z}{\partial x} = \frac{\partial}{\partial x} (3x^2 + 7xy - y^2) = 6x + 7y $$ Next, let's find the partial derivative of z with respect to y, denoted as \(\frac{\partial z}{\partial y}\): $$ \frac{\partial z}{\partial y} = \frac{\partial}{\partial y} (3x^2 + 7xy - y^2) = 7x - 2y $$
02

Find the second-order partial derivative of z with respect to y and x.

Now, we will find the second-order mixed partial derivative of z with respect to y first and then x, denoted as \(\frac{\partial^2 z}{\partial y \partial x}\): $$ \frac{\partial^2 z}{\partial y \partial x} = \frac{\partial}{\partial x} \left(\frac{\partial z}{\partial y}\right) = \frac{\partial}{\partial x} (7x - 2y) = 7 $$
03

Find the second-order partial derivative of z with respect to x and y.

Finally, we will find the second-order mixed partial derivative of z with respect to x first and then y, denoted as \(\frac{\partial^2 z}{\partial x \partial y}\): $$ \frac{\partial^2 z}{\partial x \partial y} = \frac{\partial}{\partial y} \left(\frac{\partial z}{\partial x}\right) = \frac{\partial}{\partial y} (6x + 7y) = 7 $$ We have found the second-order mixed partial derivatives, which are equal and both of them are equal to 7: $$\frac{\partial^2 z}{\partial y \partial x} = \frac{\partial^2 z}{\partial x \partial y} = 7$$

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Most popular questions from this chapter

$$ \text { If } z=14 x-13 y \text { state } \frac{\partial z}{\partial x} \text { and } \frac{\partial z}{\partial y} \text {. } $$

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