Problem 10 - APEX Calculus

Question

Find \(f_{x}, f_{y}, f_{x x}, f_{y y}, f_{x y}\) and \(f_{y x}\). $$ f(x, y)=y^{3}+3 x y^{2}+3 x^{2} y+x^{3} $$

Step-by-Step Solution

Answer

All derivatives are calculated: \(f_x = 3y^2 + 6xy + 3x^2\), \(f_y = 3y^2 + 6xy + 3x^2\), \(f_{xx} = 6y + 6x\), \(f_{yy} = 6y + 6x\), \(f_{xy} = 6y + 6x\), \(f_{yx} = 6y + 6x\).

Step 1: Find the Partial Derivative with respect to x

To find \(f_x\), take the partial derivative of \(f(x, y) = y^3 + 3xy^2 + 3x^2y + x^3\)\ with respect to \(x\). Treat \(y\) as a constant. Thus, we have:\[ f_x = 0 + 3y^2 + 6xy + 3x^2 = 3y^2 + 6xy + 3x^2 \]

Step 2: Find the Partial Derivative with respect to y

For \(f_y\), differentiate \(f(x, y)\) with respect to \(y\), treating \(x\) as a constant:\[ f_y = 3y^2 + 6xy + 3x^2 = 3y^2 + 6xy + 3x^2 \]

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

To find \(f_{xx}\), take the partial derivative of \(f_x = 3y^2 + 6xy + 3x^2\) with respect to \(x\):\[ f_{xx} = 0 + 6y + 6x = 6y + 6x \]

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

For \(f_{yy}\), differentiate \(f_y = 3y^2 + 6xy + 3x^2\) with respect to \(y\):\[ f_{yy} = 6y + 6x + 0 = 6y + 6x \]

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

Determine \(f_{xy}\) by differentiating \(f_x = 3y^2 + 6xy + 3x^2\) with respect to \(y\):\[ f_{xy} = 6y + 6x \]

Step 6: Find the Mixed Partial Derivative with respect to y then x

Find \(f_{yx}\) by differentiating \(f_y = 3y^2 + 6xy + 3x^2\) with respect to \(x\):\[ f_{yx} = 6y + 6x \]

Key Concepts

Second Partial Derivatives

When dealing with functions of several variables like \( f(x, y) \), second partial derivatives provide insights into the curvature of the function's graph. A second partial derivative is obtained by differentiating a first partial derivative. For example, if you start by finding \( f_x \) (the partial derivative of \( f \) with respect to \( x \)), the second partial derivative \( f_{xx} \) is found by differentiating \( f_x \) again with respect to \( x \). This gives you:

• \( f_{xx} = 6y + 6x \)

Similarly, for \( f_{yy} \), you first find \( f_y \), and then differentiate it again with respect to \( y \):

• \( f_{yy} = 6y + 6x \)

Second partial derivatives help us understand how the rate of change of the function's slope itself changes, which is crucial in contexts such as optimization and understanding concavity. This involves detecting if a function is concave or convex at a particular point. It’s fascinating to see how these derivatives can reveal so much about the structure and behavior of a function.

Mixed Partial Derivatives

Mixed partial derivatives are partial derivatives taken with respect to different variables. For instance, finding \( f_{xy} \) means differentiating \( f \) first with respect to \( x \), then with respect to \( y \), while \( f_{yx} \) represents the opposite order. According to Clairaut's theorem, if the function is continuous and has continuous partial derivatives, the mixed partial derivatives \( f_{xy} \) and \( f_{yx} \) are equal.

• In our example, \( f_{xy} = f_{yx} = 6y + 6x \).

This symmetry can simplify many calculations in multivariable calculus. It ensures consistency in how changes are observed across different variables, offering symmetry in mathematical models. Mixed partial derivatives are commonly used in fields like economics and physics to understand interactions between multiple changing variables.

Functions of Several Variables

Functions of several variables, such as \( f(x, y) = y^3 + 3xy^2 + 3x^2y + x^3 \), can seem complex at first. However, they represent relationships where the output is influenced by more than one input variable. In multivariable calculus, these functions allow us to model real-world phenomena like temperature in a room, financial markets, or physical systems.
Partial derivatives become particularly important for these functions, as they help us understand how each variable individually affects the output. This is useful when analyzing models or predicting outcomes.
By taking partial derivatives, we can focus on the change in the function due to a single variable while holding others constant. This zooms in on how sensitive outputs are to changes in specific inputs. It also lays the foundation for further studies in optimization, differential equations, and other advanced topics.
Grasping the role and computing of partial derivatives is key to unlocking the full potential of functions of several variables.