Differentials

Given a single-variable function $f(x)$, we call $d f$ and $d x$ the differentials. Using a Taylor series expansion, consider a small variation, $\Delta x$, of the argument of $f$ such that

Using

This gives

In infinitesimal form, as $\Delta x \rightarrow 0$, we have the differential

Now, for two-variable functions, $f(x, y)$, we have

where, again, the subscripts denote partial differentiation. This equation can be expressed as

which, in infinitesimal form is

where $d f$ is the differential of the function $f$. The above formula is useful in solving exact first-order ODEs

Finally, note that the formula for $\Delta f$ is a useful approximation; for instance, suppose we wanted to estimate the value of $\sqrt{(4.98)^{2}-(4.07)^{2}}$ without using a calculator. The number is obviously close to $\sqrt{5^{2}-4^{2}}=3$. We consider therefore a function $f(x, y)=\sqrt{x^{2}-y^{2}}$ and use the eqn with $\Delta x=4.98-5=-0.02$ and $\Delta y=4.07-4=0.07$. The partial derivatives $f_{x}$ and $f_{y}$ are given as

which, when evaluated at $(5,4)$ give $f_{x}(5,4)=5 / 3$ and $f_{y}(5,4)=-4 / 3$. The equation therefore gives,

or

Stationary points

Finally, we briefly note that the notion of stationary points extends to multivariable functions as well. Recall that in the single-variable case, a point $x=a$ is a stationary point of $f(x)$ if $f^{\prime}(a)=0$. For a function of two variables $f(x, y)$, the point $(a, b)$ is a stationary point of $f(x, y)$ if both partial derivatives vanish at $(a, b)$ i.e.

Multivariable functions are discussed in a lot more detail in multivariable functions