DIFFERENTIATION 03
Differentials: \(dx\) and \(dy\)
Differentials describe small linear changes and connect derivative notation with practical calculations.
The main idea
If \(y=f(x)\), then the derivative is
Once the derivative is known, the differential of \(y\) is defined by
Here \(dx\) represents a small chosen change in the input, while \(dy\) is the corresponding linear change predicted by the derivative.
1. From finite changes to differentials
For a finite change \(\Delta x\), the exact change in the function is
\[\Delta y=f(x+\Delta x)-f(x).\]For small \(\Delta x\), differentiability gives the approximation
\[\Delta y\approx f'(x)\Delta x.\]The differential notation records the linear part of this change:
\[dy=f'(x)dx.\]2. Example
Let \(y=x^2\). Then
\[\frac{dy}{dx}=2x.\]Therefore
At \(x=3\), if \(dx=0.01\), then
\[dy=2(3)(0.01)=0.06.\]So the derivative predicts a change of approximately \(0.06\) in \(y\).
3. Differential versus exact finite change
For the same example, the exact finite change is
\[\Delta y=(3.01)^2-3^2=0.0601.\]The differential gives
\[dy=0.06.\]Thus \(dy\) is the linear approximation to the actual finite change \(\Delta y\).
4. Is \(dy/dx\) a fraction?
The derivative \(dy/dx\) is defined through a limit, not originally as an ordinary fraction. However, after differentials are defined by \(dy=f'(x)dx\), the notation has a legitimate differential interpretation.
This explains why some fraction-like manipulations are valid when they are supported by rules such as the chain rule or separation of variables.
5. The chain rule
If \(y=f(u)\) and \(u=g(x)\), then
\[dy=\frac{dy}{du}\,du\]and
\[du=\frac{du}{dx}\,dx.\]Substituting gives
\[dy=\frac{dy}{du}\frac{du}{dx}\,dx,\]so
6. Why differentials are useful
Differentials are useful for linear approximation, error estimation, differential equations and later multivariable and stochastic calculus.
7. The central idea
The derivative gives the local rate of change; the differential uses that rate to describe a small linear change.