June 10, 2018 Sunday
Bedtime Story
Notation for Differentiation
Last night I left you with a dot.
In differential calculus, very surprisingly
even after so many years, there is no standard notation for differentiation.
There have been several giants of
mathematics (mostly of the seventeenth and eighteenth century from Europe) and
many of them came up with their own notations for the derivative of a function.
Whatever may be the controversy between
Newton and Leibniz, of one thing we are very sure; the modern notation of
differential calculus till date is largely based on the notation devised by
Gottfried Leibniz.
Leibniz’s notation is particularly useful
when one is approaching a function such y = f(x) which is a functional
relationship between a dependent and an independent variable.
Here, in the simplest of a function, x is
the independent variable whereas y is an arbitrary output or the dependent
variable, since its value will depend on the input x.
In a typical graphical representation of a
function, the horizontal axis represents the independent variable whereas the
vertical axis represents the dependent variable.
In Leibniz notation, the derivative for the
simple function is written as dy/dx.
Higher derivatives are written as d2y/dx2.
Lagrange too contributed to the notation of
differentiation and this too is still used today.
Lagrange used a prime mark to denote a
derivative and is written as f’(x).
For higher derivatives additional prime
marks are used.
Thus, second order derivative would be
denoted by double primes as f’’(x) and the third derivative is denoted by three
primes as f’’’(x).
As you see, the higher you go, the more
primes you need to add that probably gets bit awkward and unwieldy.
Euler used differential operator D and when
to our above function, would be written as Df.
Higher order functions are written as D2f,
D3f and so on.
Now let me come to the dot.
The dot notation was used by our great Newton
and is known as dot notation for differentiation where a dot is placed over the
dependent variable.
So in our given function y = f(x), where y
is a function of x, the derivative of y with respect to x is y with a dot on top of it.
Higher derivatives are then represented by
more number of dots such as y with two dots or three dots on top
As you may have noticed, Newton’s notation
compared to others is not that widely used but whenever the independent
variable is time, then the dot notation finds a niche, and this is more true
with physicists and mathematical physicists.
Moreover, its usage is generally limited to
the first and the second derivatives as most of its application in physics
suffices with just that.
Stay tuned to the voice of an average story storytelling
chimpanzee or login at http://panarrans.blogspot.com
Good night Mon Ami and my fellow cousin ape.
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Another great educator and a teacher that I am aware of is
Professor Subhashish Chattopadhyay in Bangalore, India.
While I narrate stories, Professor Subhashish an electronic
engineer and a former professor at BARC, does and teaches real mathematics and
physics.
He started the participation of Indian students at the
International Physics Olympiad.
Do visit him here:
All his books can be downloaded for free through this link:
For edutainment and English education of your children, I
recommend this large collection of Halloween Songs for Kids:
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