Differentiation

What is Differentiation in Maths

In Mathematics, Differentiation can be defined as a derivative of a function with respect to an independent variable. Differentiation, in calculus, can be applied to measure the function per unit change in the independent variable.

Let y = f(x) be a function of x. Then, the rate of change of “y” per unit change in “x” is given by:

dy / dx

If the function f(x) undergoes an infinitesimal change of ‘h’ near to any point ‘x’, then the derivative of the function is defined as

\(\begin{array}{l}\lim\limits_{h \to 0} \frac{f(x+h) – f(x)}{h}\end{array} \)

Derivative of Function As Limits

If we are given with real valued function (f) and x is a point in its domain of definition, then the derivative of function, f, is given by:

f'(a) = lim_h→0[f(x + h) – f(x)]/h

provided this limit exists.

Let us see an example here for better understanding.

Example: Find the derivative of f(x) = 2x, at x =3.

Solution: By using the above formulas, we can find,

f'(3) = lim_h→0 [f(3 + h) – f(3)]/h = lim_h→0[2(3 + h) – 2(3)]/h

f'(3) = lim_h→0 [6 + 2h – 6]/h

f'(3) = lim_h→0 2h/h

f'(3) = lim_h→0 2 = 2Notations

When a function is denoted as y = f(x), the derivative is indicated by the following notations.

D(y) or D[f(x)] is called Euler’s notation.

dy/dx is called Leibniz’s notation.

F’(x) is called Lagrange’s notation.

The meaning of differentiation is the process of determining the derivative of a function at any point.

Linear and Non-Linear Functions

Functions are generally classified into two categories under Calculus, namely:

(i) Linear functions

(ii) Non-linear functions

A linear function varies at a constant rate through its domain. Therefore, the overall rate of change of the function is the same as the rate of change of a function at any point.

However, the rate of change of function varies from point to point in the case of non-linear functions. The nature of variation is based on the nature of the function.

The rate of change of a function at a particular point is defined as a derivative of that particular function.

Differentiation Formulas

The important Differentiation formulas are given below in the table. Here, let us consider f(x) as a function and f'(x) is the derivative of the function.

If f(x) = tan (x), then f'(x) = sec2x

If f(x) = cos (x), then f'(x) = -sin x

If f(x) = sin (x), then f'(x) = cos x

If f(x) = ln(x), then f'(x) = 1/x

If f(x) = ex, then f'(x) = ex

If f(x) = xn, where n is any fraction or integer, then f'(x) = nxn-1

If f(x) = k, where k is a constant, then f'(x) = 0