Differentials

Differential equations: Separation of variables

Differentials

A very suggestive and convenient way of working with derivatives of functions emerges when we make the transition to differentials.

Differentials

A differential is an expression of the form $f(x)\, \dd g(x)$ where $f$ and $g$ are functions of $x$ . It is neither a number nor a function, but an expression indicating an "infinitely small" (infinitesimal) change depending on the change in $g(x)$ . By definition, the term satisfies the law

$f(x)\, \dd g(x) =f(x)\cdot g'(x)\,\dd x$

It is not necessary to put the $\dd$ part at the back; we agree:

$\left(\dd f(x)\right)\cdot g(x)= g(x)\, \dd f(x)$

An equation in which the terms are differentials is called an equation in differential form.

Often, the differential $1\,\dd x$ is written as $\dd x$ , and $0\,\dd x$ as $0$ .

Intuition

Consider the law

$f(x)\, \frac{\dd g(x)}{\dd x} =f(x)\cdot g'(x)$ The transition to differentials, that is, to

$\left(\dd f(x)\right)\cdot g(x)= g(x)\, \dd f(x)$ can be viewed as the multiplication of both sides of the equality by $\dd x$ . Conversely, an equality between differentials can be reduced to a differential equation by division by the appropriate differential.

The differential equation $y' =y\cdot t+t^2$ , for example, where $y$ is a function of $t$ , can be rewritten as the following equation in differential form:

$\dd y = y\cdot t\,\dd t+t^2\cdot \dd t$

We can also write $f(x)\cdot g'(x)$ as $f(x)\cdot \frac{\dd g(x) }{\dd x}$ , so the rule $f(x)\, \dd g(x) =f(x)\cdot g'(x)\,\dd x$ with $f(x) = 1$ says the same as

$\dd g(x) =\frac{\dd g(x)}{\dd x}\,\dd x$ This can be interpreted as "factors $\dd x$ in numerator and denominator cancel each other out." When we replace $g(x)$ by $y$ , this becomes even clearer: $\dd y = \frac{\dd y}{\dd x} \,\dd x$ .

The expression $f(x)\,\dd x$ behind the integral sign $\int$ in $\int f(x)\,\dd x$ is a differential.

This dual use of $0$ as the usual $0$ and as $0\,\dd x$ does not lead to confusion, because in general you cannot equate differentials to constants or functions. The $0$ in an expression like $\dd F(x,y)=0$ , for instance, must be interpreted as a differential.

By use of differentials the differentiation laws are simple to formulate:

Rules for differentiation in terms of differentials

product rule: $\dd\left(f(x)\cdot g(x) \right)= \left( \dd f(x)\right)\cdot g(x)+f(x)\cdot\dd\left( g(x) \right)$
extended summation rule: $\dd\left(a\cdot f(x)+b\cdot g(x) \right)= a\,\dd f(x)+b\,\dd g(x)$
chain rule: $\dd f\left((g(x) \right) = f'\left( g(x)\right)\,\dd g(x)$
quotient rule: $\dd \dfrac{f(x)}{g(x)} = \frac{\dd f(x)}{g(x)} -\frac{ f(x)\,\dd g(x)}{g(x)^2}$

On the right hand side of the product rule we placed $\dd\left(f(x)\right)$ before $g(x)$ .

Implicit differentiation is also easy to formulate in terms of differentials:

Implicit differentiation in terms of differentials

Let $F(x,y)$ be a function rule in $x$ and $y$ . If $C$ is a constant and $F(x,y) = C$ , then we have $\dd F(x,y)=0$ .

By way of example, we consider the circle around the origin with radius $1$ in $x,y$ -plane. This is determined by the equation

$x^2+y^2=1$

Left and right of the equality sign, we take the differential, and find $2x\,\dd x+2y\,\dd y = 0$ , which, after division by $2$ , gives

$x\,\dd x+y\,\dd y = 0$

Each circle around the origin is a solution of this equation in differential form.

Also integrals can be formulated in terms of differentials:

Integration in terms of differentials

If $f(x)\,\dd x=\dd\, F(x)$ , then $F(x) =\int f(x)\,\dd x$ .

The differential equation

$\frac{\dd}{\dd t} y=3t^2$ is equivalent to the following equation in differential form (multiply the left and right by $\dd t$ ):

$\dd y=3t^2\,\dd t$ The right hand side of this equation can also be written as $\dd(t^3)$ because $t^3$ is an antiderivative $3t^2$ . So we have an equality between two differentials:

$\dd y=\dd(t^3)$ As a consequence, the functions behind the d's are equal to each other up to constant:

$y=\int \,\dd y=\int\,\dd(t^3) = t^3+C$ for a certain constant $C$ .

More generally, if $f(x)\,\dd g(x)=\dd\, F(x)$ , then $F(x) =\int f(x)\,g'(x)\dd x$ .

The constants of integration left and right of the equality sign give no new solutions and may therefore be accommodated in a single integration constant. In the formula $F(x) =\int f(x)\,\dd x$ this happened implicitly, as the integral on the right hand side indicates a function up to a constant.