Differentiation of Algebraic Functions 1

CONTENT

1. Meaning of differentiation/derived function.
2. Differentiation from the first principle.
3. Standard derivatives of some basic functions.

MEANING OF DIFFERENTIATION/DERIVED FUNCTION

The process of finding the differential coefficient of a function is called differentiation. Differentiation deals with the measure of the rate of change in a particular function when some quantities in the function is either increased or decreased. For example, given the function \(y = f(x) \), a change in \( x\) will produce a corresponding change in \(y\). When \(y\) is increased, \(x\) is bound to increase in proportion and vice versa. Note: The reverse of differentiation is integration.

DIFFERENTIATION FROM THE FIRST PRINCIPLE

The method of finding the derivative of a function from definition is called differentiation from the first principle.  Note: A change in \(x\) to  \(x +  Δx \) produces a corresponding change in \(y \)  to \(y +  Δy \).

Example 1:

Differentiate the following from the first principle

(a) \(y = 2x + 5\)

(b) \(y = x^2 \)

SOLUTION

(a) \(y = 2x + 5\)

Take increment in both \(x\) and \(y\).