Chapter 12.6, Problem 1P

What is the difference between partial differentials and ordinary differentials?

To determine

The difference between partial differentials and ordinary differentials.

Explanation of Solution

In ordinary differentials, the total derivative of the function is depends on a single variable alone.

For example:

Consider the function $z$ depends on the variable $x$.

$z=z\left(x\right)$

The total derivative of function $z$ is expressed as follows.

$\begin{array}{c}dz=\frac{d}{dx}\left(z\right)\\ =\frac{dz}{dx}\end{array}$

In partial differentials, the total derivative of the function is depends on two variables.

For example:

Consider the function $z$ depends on the variables $x$ and $y$.

$z=z\left(x,y\right)$

The total derivative of function $z$ is expressed by partial derivative of each variables as follows.

$\begin{array}{c}dz={\left[\frac{\partial }{\partial x}\left(z\right)\right]}_{y}dx+{\left[\frac{\partial }{\partial y}\left(z\right)\right]}_{x}dy\\ ={\left(\frac{\partial z}{\partial x}\right)}_{y}dx+{\left(\frac{\partial x}{\partial y}\right)}_{x}dy\end{array}$

Here, the function $z$ is differentiated with respect to the variable $x$ by keeping the variable $y$ as constant and then the function $z$ is differentiated with respect to the variable $y$ by keeping the variable $x$ as constant.