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:

\[\dd y = y\cdot t\,\dd t+t^2\cdot \dd t\]

FractionsWe 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#.

ZeroThe 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.