Differentiation

Differentiation is a scientific mathematical concept of calculus. It can also be called as Derivative mathematically. It is most applicable mathematical concept just like trigonometry in several scientific applications especially plays significant role in industrial applications. In this article, you are going know the mathematical definition of differentiation with graphical presentation.

It is a rate of change of a particular function at one particular point by comparing with the same function’s rate at another point. In other words, the difference of performance rates of a function at two different points can also be defined Differentiation.

Mathematical Explanation & Proof of Differentiation

Assume x is an independent variable of differentiation and f(x) is a function which should be differentiated mathematically. Now, analyze this function by drawing a graph by taking differential element (x) on X Axis and corresponding f(x) values on Y Axis which means y = f(x).

Slope of the Straight-line AB
=
y - b
x - a

However, as we already know y = f(x) which also gives b = f(a). Substitute y = f(x) and b = f(a) in the mathematical formula of slope of the straight line AB.

Slope of the Straight-line AB
=
f(x) - f(a)
x - a

If point B reaches A (B → A) then the chord (straight line) become tangent of the curve. Similarly, position of x reaches to a (x → a). At that point slope of the straight line becomes slope of the tangent.

Slope of Tangent = lim
x → a
Slope of the Straight-line AB
=
lim
x → a
f(x) - f(a)
x - a

The slope of the tangent of a function f(x) at x = a is known differentiation of function f(x). Differentiation of f(x) is represented with f '(x), D.f(x), d.f(x)/dx and f1(x).

d.f(x)
dx
=
 Slope of Tangent = lim
x → a
Slope of the Straight-line AB
=
lim
x → a
f(x) - f(a)
x - a

Simply, we can write this mathematical definition of differentiation.

d.f(x)
dx
=
lim
x → a
f(x) - f(a)
x - a

Case: 1

Assume x - a = h » a = x - h

According to mathematical explanation, as we already know x reaches to a (x → a) therefore the difference between x and a; should be zero (x - a → 0). However, as per our assumption the difference between x and a is h. So that h → 0. Just substitute a = x – h and h → 0 instead of x → a in the mathematical definition of differentiation.

d.f(x)
dx
=
lim
h → 0
f(x) - f(x-h)
- h
-----------------> (1)

Case: 2

Assume a - x = h » a = x + h

As per fundamental definition of differentiation, we knew x reaches to a (x → a) therefore the difference between x and a; should be zero (x - a → 0). Similarly, the differentiation between a and x should also be zero (a - x → 0). However, as per our assumption the difference between a and x is h. Therefore h → 0. Substitute a = x + h and h → 0 instead of x → a in the mathematical definition of differentiation.

d.f(x)
dx
=
lim
x → a
f(x) - f(x+h)
- h

For our convenience, we can write this function as stated below by taking –VE value common from the numerator.

d.f(x)
dx
=
lim
x → a
- [f(x+h) - f(x) ]
- h

This negative sign can be eliminated because of the same sign in numerator and denominator.

d.f(x)
dx
=
lim
x → a
[f(x+h) - f(x) ]
h
---------------> (2)

You can use either (1) or (2) fundamental to determine the derivative of any function.