Differentials and integrals: Differentials
Differential of a function
The previous example can be generalized as follows: Suppose the function \(y=f(t)\) is differentiable in the point \(t\), that is, has a non-vertical tangent line in this point. For a small increase \({\vartriangle}t\) of \(t\), one can estimate the increase \({\vartriangle}y=f(t+{\vartriangle}t)-f(t)\) well by the following formula: \[{\vartriangle}y\approx f'(t)\cdot {\vartriangle}t\] The closer in the neighbourhood of \(t\), that is, the smaller \({\vartriangle}t\), the better is the estimated value of the function. The underlying idea of this formula is that the graph of a smooth function is in practice almost a straight line when one sufficiently zoomes in. For negligible changes, denoted with \(dy\) and \(dt\), the following statement can be written down:
\[\text{If }y=f(t), \text{ then }dy=f'(t)\,dt\] The right-hand side \(f'(t)\,dt\) is called the differential of f and one uses for this the notations \(dy\), \(df\) and \(d\bigl(f(t)\bigr)\).
The differential \(f\) therefore depends on the time \(t\) and the infinitesimal change \(dt\). If \(dy\) is denoted as \(df\), the relationship between differentials and derivatives is easily made: the quotient of the differentials \(df\) and \(dt\), also known as differential quotient is equal to the derivative \(f'(t)\) in \(t\). Hence the mixed use of \(y'\) and \(\displaystyle\frac{dy}{dt}\) for the derivative function in this instructional material.
It is thus clear that a differential equation, that is, an equation in which besides a yet unknown function also one or more derivatives of that function are present can also be written as an equation between differentials: for example, the differential equation \(\displaystyle\frac{dy}{dt}=y\) can be written as a relationship between differentials as \(dy=y\,dt\). More generally, the differential equation \(\displaystyle\frac{dy}{dt}=\varphi(t,y)\) with \(\varphi\) a function of two variables, can be rewritten in terms of differentials as \(dy=\varphi(t,y)\,dt\).
Example of the use of differentials Even if the two notations for a differential equation are equivalent, in practice the language of differentials appear to be more convenient for formulating and solving a differential equation. The following example of the derivation of the formula for the area of a circle of given radius may illustrate this. Consider the area \(A\) of a circle as a quantity that depends on the radius \(r\). Take a circle whose radius is slightly larger, say with \(r+{\vartriangle}r\). Then the area has increased a little bit, say with \({\vartriangle}A\). These small changes are related to each other via the formula \({\vartriangle}A\approx C\cdot {\vartriangle}r\) where \(C\) is the circumference of the circle with radius \(r\). After all, the circular strip with width \({\vartriangle}r\), which is added to the circle of radius \(r\), can be transformed smoothly into a rectangle with length \(C\) and width \({\vartriangle}r\) when this width is sufficiently small. For infinitesimal changes we get in this way the equation \(dA=C\,dr\). The derivative \(\displaystyle\frac{dA}{dr}\) is therefore equal to \(C\). With the knowledge of \(C=2\pi r\), we find that the derivative of \(A\) is equal to \(2\pi r\) and thus \(A=\pi r^2.\)
