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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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Partial Derivatives
Partial derivatives are an essential tool in multivariable calculus. They allow us to analyze the rate of change of a function with respect to one of several variables, while keeping the others fixed. This is like finding a derivative, but with a focus on each specific variable in a function involving more than one. For a function like \( z = 3x^2 + 7xy - y^2 \), you can think of it as a landscape, and partial derivatives help us understand how the height changes if we move in only one direction.
- When finding \( \frac{\partial z}{\partial x} \), we consider 'y' as a constant, so the expression related to 'y' does not change its value. The result is \( 6x + 7y \).
- Similarly, for \( \frac{\partial z}{\partial y} \), 'x' is treated as a constant. Here, the outcome is \( 7x - 2y \).
Mixed Partial Derivatives
Mixed partial derivatives are fascinating and important for understanding the intricacies of multivariable functions. They involve taking the partial derivative of a function with respect to one variable and then taking the derivative of the result with respect to another variable.
Consider the function \( z = 3x^2 + 7xy - y^2 \). To find **mixed partial derivatives**, we follow two steps:
Consider the function \( z = 3x^2 + 7xy - y^2 \). To find **mixed partial derivatives**, we follow two steps:
- First, find the partial derivative with respect to 'y', and then with respect to 'x'. This is written as \( \frac{\partial^2 z}{\partial y \partial x} \).
- Alternatively, switch the order and first take the partial derivative with respect to 'x', then 'y', noted as \( \frac{\partial^2 z}{\partial x \partial y} \).
Differentiation
Differentiation, the cornerstone of calculus, is the process of finding a derivative, which measures how a function changes as its input changes. This concept extends neatly into multivariable calculus through the use of partial and mixed derivatives.
Partial differentiation entails focusing on one variable at a time, simplifying the process of dealing with functions of several variables. By doing so, we break down complex systems into easier parts, akin to solving a puzzle one piece at a time. Differentiation effectively allows us to zoom in on the function's landscape, finding slopes and changes from one direction to the next.
Partial differentiation entails focusing on one variable at a time, simplifying the process of dealing with functions of several variables. By doing so, we break down complex systems into easier parts, akin to solving a puzzle one piece at a time. Differentiation effectively allows us to zoom in on the function's landscape, finding slopes and changes from one direction to the next.
- In the given function \( z = 3x^2 + 7xy - y^2 \), differentiation with respect to each variable helps isolate and understand how 'x' and 'y' affect 'z' when considered alone.
- It provides insight into the function's dynamics, enabling predictions about how it behaves under various conditions.