Problem 22 - Applied Calculus

Question

For Problems calculate all four second-order partial derivatives and confirm that the mixed partials are equal. $$f(x, y)=x e^{y}$$

Step-by-Step Solution

Answer

The second-order partial derivatives confirm that the mixed partials are equal: \( f_{xy} = f_{yx} = e^y \).

Step 1: Find the First Partial Derivative with respect to x

To find the first partial derivative of the function \( f(x, y) = x e^y \) with respect to \( x \), treat \( y \) as a constant. The derivative of \( x \) with respect to \( x \) is 1, hence: \[ f_x(x, y) = e^y \]

Step 2: Find the First Partial Derivative with respect to y

To find the first partial derivative of the function \( f(x, y) = x e^y \) with respect to \( y \), treat \( x \) as a constant. The derivative of \( e^y \) with respect to \( y \) is \( e^y \), so:\[ f_y(x, y) = x e^y \]

Step 3: Find the Second Partial Derivative with respect to x

Differentiate the first partial derivative \( f_x(x, y) = e^y \) again with respect to \( x \). As \( e^y \) is independent of \( x \), the derivative is:\[ f_{xx}(x, y) = 0 \]

Step 4: Find the Second Partial Derivative with respect to y

Differentiate the first partial derivative \( f_y(x, y) = x e^y \) again with respect to \( y \). The derivative of \( x e^y \) with respect to \( y \) is:\[ f_{yy}(x, y) = x e^y \]

Step 5: Find the Mixed Partial Derivative \( f_{xy} \)

Differentiate \( f_x(x, y) = e^y \) with respect to \( y \). This gives:\[ f_{xy}(x, y) = e^y \]

Step 6: Find the Mixed Partial Derivative \( f_{yx} \)

Differentiate \( f_y(x, y) = x e^y \) with respect to \( x \). This results in:\[ f_{yx}(x, y) = e^y \]

Step 7: Confirm that the Mixed Partial Derivatives are Equal

Both mixed partial derivatives \( f_{xy} \) and \( f_{yx} \) are equal to \( e^y \). Therefore, \( f_{xy} = f_{yx} \), confirming Clairaut's theorem for mixed partials.

Key Concepts

First Partial Derivative

When dealing with multivariable calculus, finding the first partial derivative is a key concept. A partial derivative shows us how a function changes if we tweak just one variable at a time, leaving the other variables constant.
For example, in our function \( f(x, y) = x e^y \), we can find the first partial derivative with respect to each variable, \( x \) and \( y \).

• Partial Derivative with respect to \( x \): Consider \( y \) constant. The derivative of \( x \, e^y \) becomes \( f_x(x, y) = e^y \), because \( e^{y} \) acts as a constant multiplier to \( x \).
• Partial Derivative with respect to \( y \): Now treat \( x \) as a constant. The derivative of \( e^y \) is still \( e^y \), but multiplied by \( x \), giving us \( f_y(x, y) = x \, e^y \).

Understanding these derivatives helps us understand the initial rates of change in multivariable functions.

Mixed Partial Derivatives

Mixed partial derivatives come into play when we differentiate a function first with respect to one variable, and then with respect to another. This process tells us how the function changes when we adjust two variables in sequence.
In the exercise, we compute two types of mixed partial derivatives for \( f(x, y) = x e^y \):

• First, \( f_{xy} \): We first take the derivative of the function with respect to \( x \), obtaining \( f_x(x, y) = e^y \). Then we differentiate this result with respect to \( y \), leading us to \( f_{xy}(x, y) = e^y \).
• Next, \( f_{yx} \): We first find the derivative with respect to \( y \), getting \( f_y(x, y) = x e^y \). Differentiating again with respect to \( x \), we achieve \( f_{yx}(x, y) = e^y \).

This symmetry in partial derivatives is a fascinating concept that leads us to Clairaut's Theorem.

Clairaut's Theorem

Clairaut's Theorem is a beautiful aspect of calculus, dealing with the interchangeability of mixed partial derivatives. According to this theorem, if your function is smooth enough (meaning it has continuous second-order derivatives), then the order of differentiation doesn't change the result.
In our example function, \( f(x, y) = x e^y \), we see this principle in action:

• Both \( f_{xy}(x, y) \) and \( f_{yx}(x, y) \) compute to \( e^y \).

This equality confirms Clairaut's Theorem, showing that the mixed partial derivatives are the same regardless of the differentiation order. Such insights are crucial when analyzing functions with multiple variables, ensuring that different paths of computation lead to the same result.