Problem 22 - Applied Calculus

Question

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

Mixed partials \( f_{xy} \) and \( f_{yx} \) are both \( e^y \), confirming they are equal.

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

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

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

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

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

Now differentiate \( f_x = e^y \) with respect to \( x \). Since \( e^y \) is a constant with respect to \( x \), its derivative is:\[ f_{xx} = 0 \]

Step 4: Find the Mixed Partial Derivative with Respect to y then x

Differentiate \( f_y = x e^y \) with respect to \( x \). Since \( e^y \) is treated as a constant in differentiation with respect to \( x \), the derivative is:\[ f_{yx} = e^y \]

Step 5: Find the Mixed Partial Derivative with Respect to x then y

Differentiate \( f_x = e^y \) with respect to \( y \). Since \( x \) is constant for this partial derivative, you have:\[ f_{xy} = e^y \]

Step 6: Find the Second Partial Derivative with Respect to y

Finally, differentiate \( f_y = x e^y \) with respect to \( y \). This requires applying the derivative to \( x e^y \), yielding:\[ f_{yy} = x e^y \]

Step 7: Confirm the Mixed Partials are Equal

Compare the mixed partial derivatives \( f_{xy} \) and \( f_{yx} \). Both are equal to \( e^y \), confirming the equality of mixed partials.

Key Concepts

Mixed partial derivatives

Mixed partial derivatives involve taking the derivative of a function with respect to more than one variable in succession. Here, for the given function \( f(x, y) = x e^y \), mixed partial derivatives consider the order of differentiating with respect to \( x \) and \( y \).

This means you first differentiate with respect to one variable and then take the result and differentiate again with respect to the other variable. The function has two mixed derivatives:

• \( f_{xy} \) - differentiate first with respect to \( x \) and then with respect to \( y \).
• \( f_{yx} \) - differentiate first with respect to \( y \) and then with respect to \( x \).

Both of these steps yield the same results, demonstrating an important property related to the equality of mixed partial derivatives, which is explored further in later sections.

Partial derivative with respect to x

To find the partial derivative with respect to \( x \), you treat all other variables as constants. Here, in the function \( f(x, y) = x e^y \), treat \( y \) as a constant.

The first step is to differentiate \( f \) with respect to \( x \). Since \( e^y \) acts like a constant multiplier, differentiating gives:

• \( f_x = e^y \) - this is because the derivative of \( x \) with respect to itself is 1, and the constant \( e^y \) stays as it is.

The next step is to find the second partial derivative, \( f_{xx} \), by differentiating \( f_x \) with respect to \( x \) again. Here, since \( e^y \) is constant with respect to \( x \), its derivative is:

• \( f_{xx} = 0 \)

Partial derivative with respect to y

Finding the partial derivative with respect to \( y \) involves treating \( x \) as a constant. In the case of \( f(x, y) = x e^y \), differentiate with respect to \( y \).

Here, \( x \) operates as a constant multiplier to \( e^y \), so when differentiating, apply the exponential rule:

• \( f_y = x e^y \)

To find the second partial derivative, \( f_{yy} \), differentiate \( f_y \) with respect to \( y \). Using the derivative of the exponential function yields:

• \( f_{yy} = x e^y \)

This reaffirms that both \( x \) and \( e^y \) are consistently involved when deriving with respect to \( y \) again.

Equal mixed partials

The concept of equal mixed partials is a fundamental property in mathematics, known as Clairaut's theorem or Schwarz's theorem. It states that if a function is continuous and all partial derivatives up to the required order exist, the mixed derivatives are equal.

For the function \( f(x, y) = x e^y \), after solving for \( f_{xy} \) and \( f_{yx} \), both equal \( e^y \). This confirms the equality:

• \( f_{xy} = e^y \)
• \( f_{yx} = e^y \)

This agreement in mixed partials not only satisfies theoretical expectations but also provides consistency in calculations. Equal mixed partials assure that regardless of the differentiation order, the results remain the same.