Problem 14 - APEX Calculus

Question

Find \(f_{x}, f_{y}, f_{x x}, f_{y y}, f_{x y}\) and \(f_{y x}\). $$ f(x, y)=e^{x+2 y} $$

Step-by-Step Solution

Answer

\( f_x = e^{x+2y}, f_y = 2e^{x+2y}, f_{xx} = e^{x+2y}, f_{yy} = 4e^{x+2y}, f_{xy} = f_{yx} = 2e^{x+2y} \).

Step 1: Find the Partial Derivative of f with respect to x, f_x

To find the partial derivative of the function \( f(x, y) = e^{x+2y} \) with respect to \( x \), we treat \( y \) as a constant. The derivative of \( e^{x+2y} \) with respect to \( x \) is just \( e^{x+2y} \), so we have: \(\( f_x = e^{x+2y} \).\)

Step 2: Find the Partial Derivative of f with respect to y, f_y

To find the partial derivative with respect to \( y \), treat \( x \) as a constant. The derivative of \( e^{x+2y} \) with respect to \( y \) is \( 2e^{x+2y} \) because you differentiate \( 2y \), which yields 2. Thus, \(f_y = 2e^{x+2y}.\)

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

Find the derivative of \( f_x = e^{x+2y} \) with respect to \( x \). Since \( e^{x+2y} \) is its own derivative, we get: \(f_{xx} = e^{x+2y}.\)

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

Find the derivative of \( f_y = 2e^{x+2y} \) with respect to \( y \). Differentiating \( 2e^{x+2y} \) with respect to \( y \) produces another factor of 2: \(f_{yy} = 4e^{x+2y}.\)

Step 5: Find the Mixed Partial Derivative, f_xy

Find the derivative of \( f_x = e^{x+2y} \) with respect to \( y \). Differentiating with respect to \( y \) gives \( 2e^{x+2y} \): \(f_{xy} = 2e^{x+2y}.\)

Step 6: Find the Mixed Partial Derivative, f_yx

Find the derivative of \( f_y = 2e^{x+2y} \) with respect to \( x \). This is the same as \( f_{xy} \): \(f_{yx} = 2e^{x+2y}.\) By Clairaut's Theorem, \( f_{xy} = f_{yx} \).

Key Concepts

Second Partial Derivatives

Second partial derivatives are derivatives of derivatives. Imagine you've already calculated the first partial derivative with respect to one variable; now you differentiate it again, either with respect to the same variable or a different one.
For the function given, after finding the first partial derivatives,

• \(f_x\) and \(f_y\),

you move on to second partial derivatives.

• \(f_{xx}\) is the derivative of \(f_x = e^{x+2y}\) with respect to \(x\) again. This treats \(y\) as a constant and yields \(e^{x+2y}\).
• Similarly, \(f_{yy}\) is the derivative of \(f_y = 2e^{x+2y}\) with respect to \(y\) again, resulting in another factor of 2, making it \(4e^{x+2y}\).

These second partials help describe how the function curves in different directions, giving more insight into the behavior of the surface described by the function.

Clairaut's Theorem

Clairaut's Theorem, a vital result in calculus, gives us an essential rule about mixed partial derivatives. It tells us that if the mixed partial derivatives of a function are continuous at a point, then the order of differentiation doesn't matter.
This means:

• \(f_{xy}\) is equal to \(f_{yx}\).

In our solution, both derivatives are calculated:

• When finding \(f_{xy}\), we differentiate \(f_x = e^{x+2y}\) with respect to \(y\), leading to \(2e^{x+2y}\).
• Likewise, \(f_{yx}\) involves differentiating \(f_y = 2e^{x+2y}\) with respect to \(x\), which also results in \(2e^{x+2y}\).

This agreement between \(f_{xy}\) and \(f_{yx}\) illustrates Clairaut's Theorem beautifully.

Mixed Partial Derivatives

Mixed partial derivatives are derivatives taken with respect to different variables at each step. In simpler terms, after deriving once with one variable, a mixed partial derivative continues the process with another variable.
In our example, after computing the first partial derivatives:

• We calculated \(f_{xy}\), the derivative of \(f_x\) with respect to \(y\), resulting in \(2e^{x+2y}\).
• We also computed \(f_{yx}\), the derivative of \(f_y\) with respect to \(x\), which resulted in the same outcome, \(2e^{x+2y}\).

These derivatives provide a glimpse into how changes along one axis affect the slope along another. Mixed partials are incredibly useful in fields like physics or economics, where multiple variables often interact in dynamic ways.