Consider a differentiable function \(f\) of two variables \(z=f(x,y)\). If \(x\) changes to \(x+\Delta x\) and \(y\) changes to \(y+\Delta y\), the increment of \(f\), \(\Delta f\), from the definition of differentiability , can be written as:
\begin{align}
\Delta f&= f(x+\Delta x,y+\Delta y)-f(x,y)\\
& =f_x(x,y)\Delta x+f_y(x,y)\Delta y+\sqrt{(\Delta x)^2+(\Delta y)^2}\ \ \varepsilon(\Delta x,\Delta y).\tag{i}
\end{align}
We take the linear part of \(\Delta f\) and call them the differential of \(f\). The differential part of \(f\) is denoted by \(df\) or \(dz\):
\[\begin{aligned} \label{Eq:df0}
dz=df=&\frac{\partial f}{\partial x}\Delta x+\frac{\partial f}{\partial y}\Delta y\\
=&f_x(x,y)\Delta x+f_y(x,y)\Delta y.
\end{aligned}\]
- Note that \(df\) is a function of four variables: \(x, y,\Delta x\), and \(\Delta y\). To emphasize on that, we may write it as \(df(x,y,\Delta x,\Delta y)\).
If \(x\) and \(y\) are independent variables, from Equation (i) we have:
\[dx=\underbrace{\frac{\partial x}{\partial x}}_{=1}\Delta x+\underbrace{\frac{\partial x}{\partial y}}_{=0}\Delta y=\Delta x,\]
and
\[dy=\underbrace{\frac{\partial y}{\partial x}}_{=0}\Delta x+\underbrace{\frac{\partial y}{\partial y}}_{=1}\Delta y=\Delta y.\]
Then Equation (i) takes the form:
\[df=\frac{\partial f}{\partial x} dx+\frac{\partial f}{\partial y} dy=f_x(x,y) dx+f_y(x,y) dy.\]
The above expression is sometimes called the total differential of \(f(x,y)\).
Definition 1. If \(z=f(x,y)\) is a differentiable function at \((x,y)\), the total differential of \(f\) is the function \(df\) defined by:
\[df(x,y,dx,dy)=\frac{\partial f}{\partial x} dx+\frac{\partial f}{\partial y} dy=f_x(x,y)dx+f_y(x,y) dy\]
When u = f(x, y, z):
Obviously, we can extend these methods and results to functions of any number of variables. For example, if \(u=f(x,y,z)\), then
\[ \bbox[#F2F2F2,5px,border:2px solid black]{du=df=\frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy+\frac{\partial f}{\partial z} dz.}\]
In general, when y = f(x1, x2, …, xn):
In this case,
\[ \bbox[#F2F2F2,5px,border:2px solid black]{dy=df=\frac{\partial f}{\partial x_1}dx_1+\cdots+\frac{\partial f}{\partial x_n} dx_n=\sum_{i=1}^n\frac{\partial f}{\partial x_i} dx_i.}\]