91Ó°ÊÓ

In mathematics, linear functions are incredibly simple yet powerful. They take the form \( f(x, y) = ax + by + c \), characterized by constants \( a \), \( b \), and \( c \). These represent the weights of the variables \( x \) and \( y \), with \( c \) being a constant term.
Linear functions form straight lines when graphed, with their slope defined by the coefficients \( a \) and \( b \). In a graphical sense, changing \( x \) and \( y \) leads to predictable increases or decreases along a plane, following the linear path.

• These functions do not exhibit curvature, making them easy to analyze.
• They are foundational in calculus and linear algebra, providing a base for more complex functions.

Recognizing a linear function is easy – just look for the highest power of \( x \) and \( y \) to be one.

Mixed partial derivatives arise when a function is differentiated with respect to one variable and then another. For example, taking the derivative of \( f(x, y) = ax + by + c \) first with respect to \( x \), and then with respect to \( y \), gives you the mixed partial derivative \( f_{xy} \).
In the context of our linear function, mixed partial derivatives are zero, since there are no terms in the function that involve two or more variables multiplied together. To clarify:

• \( f_{xy} \) involves differentiating \( f_x = a \) with respect to \( y \), resulting in 0 because \( a \) is constant with respect to \( y \).
• Similarly, \( f_{yx} \) involves differentiating \( f_y = b \) with respect to \( x \), also yielding 0.

By Clairaut's theorem, if the function and its partial derivatives are continuous, \( f_{xy} = f_{yx} \). This is a crucial property that simplifies many calculations in calculus.

First partial derivatives of a function represent the rate of change of that function with respect to one of its variables, keeping the other variables constant. For a linear function like \( f(x, y) = ax + by + c \), finding the first partial derivatives is straightforward.
The derivative with respect to \( x \), noted as \( f_x \), simplifies to \( a \), since \( by + c \) are treated as constants.

• \( f_x = \frac{\partial}{\partial x}(ax + by + c) = a \)

On the other hand, the derivative with respect to \( y \), denoted \( f_y \), simplifies to \( b \) for the same reasons.

• \( f_y = \frac{\partial}{\partial y}(ax + by + c) = b \)

First partial derivatives are critical as they indicate the slope or gradient in the direction of each variable. This understanding is essential for more complex calculus applications, like optimization and modeling changes in multivariable functions.