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Magazine Chapter 6: Problem 34
Find \(f_{x x}, f_{x y}, f_{y x},\) and \(f_{y y}\) (Remember, \(f_{y x}\) means to differentiate with respect to \(y\) and then with respect to \(x\).) $$f(x, y)=3 x+5 y$$
Short Answer
Expert verified
All second derivatives are 0: \(f_{xx} = 0\), \(f_{xy} = 0\), \(f_{yx} = 0\), \(f_{yy} = 0\).
Step by step solution
01
- Compute the first partial derivative with respect to x
Given the function \(f(x, y) = 3x + 5y\), to find \(f_x\) (the partial derivative of \(f\) with respect to \(x\)), differentiate with respect to \(x\) while treating \(y\) as a constant. \( f_x = \frac{\partial}{\partial x} (3x + 5y) = 3 \).
02
- Compute the second partial derivative with respect to x
To find \(f_{xx}\), or the second partial derivative of \(f\) with respect to \(x\), differentiate \(f_x\) with respect to \(x\):\( f_{xx} = \frac{\partial}{\partial x}(3) = 0 \).
03
- Compute the first partial derivative with respect to y
To find \(f_y\) (the partial derivative of \(f\) with respect to \(y\)), differentiate with respect to \(y\) while treating \(x\) as a constant. \( f_y = \frac{\partial}{\partial y}(3x + 5y) = 5 \).
04
- Compute the second partial derivative with respect to y
To find \(f_{yy}\), or the second partial derivative of \(f\) with respect to \(y\), differentiate \(f_y\) with respect to \(y\):\( f_{yy} = \frac{\partial}{\partial y}(5) = 0 \).
05
- Compute the mixed partial derivative \(f_{xy}\)
To find \(f_{xy}\), first find \(f_x = 3\) (already computed in Step 1). Then, differentiate \(f_x\) with respect to \(y\):\( f_{xy} = \frac{\partial}{\partial y}(3) = 0 \).
06
- Compute the mixed partial derivative \(f_{yx}\)
To find \(f_{yx}\), first find \(f_y = 5\) (already computed in Step 3). Then, differentiate \(f_y\) with respect to \(x\):\( f_{yx} = \frac{\partial}{\partial x}(5) = 0 \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
second partial derivatives
Second partial derivatives are simply the partial derivatives of the partial derivatives. To compute them, you differentiate the function twice, first with respect to one variable and then possibly again with respect to the same or a different variable. For instance, if we have a function \(f(x, y) = 3x + 5y\), and we need to find \(f_{xx}\), the second partial derivative with respect to \(x\), we first differentiate with respect to \(x\):
Similarly, to find \(f_{yy}\), differentiate first with respect to \(y\):
- First Partial Derivative with respect to \(x\): \( f_x = \frac{\frac{df}{\frac{dx}}}{dx} (3x + 5y) = 3 \).
- Second Partial Derivative with respect to \(x\): \( f_{xx} = \frac{\frac{df}{\frac{df_x}}}{dx}(3) = 0 \).
Similarly, to find \(f_{yy}\), differentiate first with respect to \(y\):
- First Partial Derivative with respect to \(y\): \( f_y = \frac{\frac{df}{\frac{dy}}}{dy}(3x + 5y) = 5 \).
- Second Partial Derivative with respect to \(y\): \( f_{yy} = \frac{\frac{d}{\frac{dy}} (5)}{dy} = 0 \).
mixed partial derivatives
Mixed partial derivatives involve differentiating a function first with respect to one variable and then with respect to another. These are often represented as \(f_{xy}\) or \(f_{yx}\).
Let's go through an example using \(f(x, y) = 3x + 5y\). To find \(f_{xy}\), differentiate \(f\) first with respect to \(x\) to get \(f_x\), then with respect to \(y\).
For \(f_{yx}\), we reverse the order:
Let's go through an example using \(f(x, y) = 3x + 5y\). To find \(f_{xy}\), differentiate \(f\) first with respect to \(x\) to get \(f_x\), then with respect to \(y\).
- First Partial Derivative with respect to \(x\): \( f_x = \frac{d(3x + 5y)}{dx} = 3 \).
- Mixed Partial Derivative \(f_{xy}\): \( f_{xy} = \frac{d(3)}{dy} = 0 \).
For \(f_{yx}\), we reverse the order:
- First Partial Derivative with respect to \(y\): \( f_y = \frac{d(3x + 5y)}{dy} = 5 \).
- Mixed Partial Derivative \(f_{yx}\): \( f_{yx} = \frac{d(5)}{dx} = 0 \).
differentiation
Differentiation is the process of finding the derivative of a function, which indicates how the function changes as its input changes. In terms of partial derivatives, it involves differentiating with respect to just one of multiple variables, treating others as constants.
For example, given \(f(x, y) = 3x + 5y\):
Partial differentiation helps in understanding how a function behaves when one variable changes, independent of other variables. By mastering differentiation, you lay the groundwork for more complex topics like optimization and differential equations in multivariable calculus. Each step, carefully analyzed, builds intuitive understanding and proficiency in calculus.
For example, given \(f(x, y) = 3x + 5y\):
- Differentiate with respect to \(x\), treating \(y\) as a constant: \( f_x = \frac{d(3x + 5y)}{dx} = 3 \).
- Differentiate with respect to \(y\), treating \(x\) as a constant: \( f_y = \frac{d(3x + 5y)}{dy} = 5 \).
Partial differentiation helps in understanding how a function behaves when one variable changes, independent of other variables. By mastering differentiation, you lay the groundwork for more complex topics like optimization and differential equations in multivariable calculus. Each step, carefully analyzed, builds intuitive understanding and proficiency in calculus.
