91Ó°ÊÓ

Second order derivatives are essentially the derivatives of derivatives. For a function of two variables such as \( h(x, y) = x e^y + y + 1 \), the process involves differentiating the first order derivatives again with respect to the variables \( x \) and \( y \). The second order partial derivatives provide insights into the function's curvature along different axes.
Let's break it down:

• The second order partial derivative with respect to \( x \), \( \frac{\partial^2 h}{\partial x^2} \), involves differentiating the first derivative \( \frac{\partial h}{\partial x} = e^y \) again with respect to \( x \). Since \( e^y \) is independent of \( x \), the result is 0.
• The second order partial derivative with respect to \( y \), \( \frac{\partial^2 h}{\partial y^2} \), requires differentiating \( \frac{\partial h}{\partial y} = x e^y + 1 \) again with respect to \( y \). Here, the derivative of \( x e^y \) is \( x e^y \), while 1 is a constant with a derivative of 0, resulting in \( x e^y \).

Understanding second order derivatives helps in analyzing concavity and determining the nature of function behavior, which is crucial in maximum and minimum problems in calculus.

Mixed partial derivatives occur when we take partial derivatives of a function successively with respect to different variables. For functions like \( h(x, y) = x e^y + y + 1 \), these derivatives can tell us about how the function changes when both variables are altered together.
In this exercise:

• The mixed partial derivative \( \frac{\partial^2 h}{\partial x \partial y} \) is found by differentiating \( \frac{\partial h}{\partial x} = e^y \) with respect to \( y \). This results in \( e^y \) as the derivative of \( e^y \) with respect to \( y \) is \( e^y \).
• The reverse order, \( \frac{\partial^2 h}{\partial y \partial x} \), involves differentiating \( \frac{\partial h}{\partial y} = x e^y + 1 \) with respect to \( x \), which gives \( e^y \) as \( e^y \) is the coefficient of \( x \) in the term \( x e^y \).

A fascinating property of mixed partial derivatives, observed in most smooth functions, is their equality: \( \frac{\partial^2 h}{\partial x \partial y} = \frac{\partial^2 h}{\partial y \partial x} \). This principle, known as Clairaut's theorem, holds true when the mixed partial derivatives are continuous. It simplifies calculations and ensures consistency in evaluation across different variable orders.

Partial derivatives are fundamental in dealing with functions of multiple variables because they allow us to see how a function changes as one specific variable changes, while keeping others constant. Taking a closer look at the function \( h(x, y) = x e^y + y + 1 \), we focus on how each term reacts when isolated variable changes are made.
Here's the process for finding the first order partial derivatives:

• For \( \frac{\partial h}{\partial x} \), consider \( y \) constant. The derivative of \( x e^y \) with respect to \( x \) is \( e^y \), while \( y + 1 \) gives 0, making \( \frac{\partial h}{\partial x} = e^y \).
• For \( \frac{\partial h}{\partial y} \), fix \( x \) and differentiate. \( x e^y \) with respect to \( y \) becomes \( x e^y \) and \( y + 1 \)'s derivative is simply 1, resulting in \( \frac{\partial h}{\partial y} = x e^y + 1 \).

These first order partial derivatives shed light on the slope and elasticity of the function's surface along particular directions. They are essential tools in optimization and in understanding the geometry of multi-variable functions.