Infinitesimals and Differentials.





In the chain rule we have come across the following relation:

\frac{dy}{dx} = \frac{dy}{du}\times\frac{du}{dx}

Which gives us impression that \frac{dy}{dx} is a fraction of “dy” and “dx”

And in right hand side of above relation we can cancel “du” in the numerator and denominator to get \frac{dy}{dx} of the left hand side.

Actually in the early days Leibnitz and others treated \frac{dy}{dx} as a fraction of “dy” and “dx” or delta y and delta x which are

called infinitesimals or infinitely small numbers having basic properties of the number zero.

In fact this is the reason why British were one or two hundreds years behind than Europe in development of calculus , Europeans regarded dy and dx as infinitesimals

but Britishers didn’t.

As long as a infinitesimals is added or multiplied with a number the infinitesimals acts as the number zero. But unlike zero when an infinitesimals is divided by

another infinitesimals the result is a derivative which can be a function or a constant. This interpretation of \frac{dy}{dx} being fallacious

, Cauchy gave a different type of interpretation and introduced the concept of an operator.

But there is some advantage in considering \frac{dy}{dx} as a fraction , So latter the concept of differential is introduced to replace the idea

infinitesimal.

Let us understand the concept of Differentials with the help of following image:

Concept of Differentials.

In the figure , Let AB be a curve given by the function y=f(x) and P be a point on the curve whose co-ordinates are (x,y) let Q be a point which is very near to P.

So , it’s co-ordinates can be written as:

 (x+\Delta x ,y+ \Delta y). Now draw the tangent PT at P which cuts the ordinate QM at T. let PR be perpendicular to QM.

We know that  f ‘ (x) or the derivative of function at x gives the gradient of the curve at the point P or x. which is also measured by:

\dfrac{TR}{PR}

So ,

\dfrac{TR}{PR} = f ‘(x)

But , PR = LM  = OM-OL=   x+\Delta x - x \Delta x

SO , TR= f ‘(x) * \Delta x

When a point moves along the curve from P to Q , there is change in \Delta x in the abscissa and

\Delta y   in the ordinate of P. But if the point moves along the tangent from P to T , we have change TR in the ordinate for the same change \Delta x in the abscissa.  This change or increment TR is called the differential  of y and is denoted by “dy” or “d f(x)”

So , dy = f ‘ (x)  . \Delta x

The corresponding change or increment  in abscissa which is PR is called the differential of x and is denoted by dx. but PR = \Delta x

So ,

dx = \Delta x

and ,

dy = f ‘ (x)  . \Delta x = f ‘ (x)  .  dx



Related posts:

  1. The Chain Rule. Chain Rule is one of the Techniques of Differentiation. The...
  2. Derivative of simple algebraic or polynomial functions. The derivative and calculations on finding derivative of simple algebraic functions...
  3. The Quotient Rule. Quotient Rule is one of the Techniques of Differentiation. The...
  4. Second and higher derivatives. Think of a function y=f(x) , and let y=f(x) be...
  5. The Product Rule. Product Rule is one of the Techniques of Differentiation. The...