THE DERIVATIVE AND DIFFERENTIATION OF ALGEBRAIC FUNCTIONS OBJECTIVES

THE DERIVATIVE AND DIFFERENTIATION OF ALGEBRAIC FUNCTIONS OBJECTIVES: • to define the derivative of a function • to find the derivative of a function by increment method (4 -step rule) • to identify the different rules of differentiation and distinguish one from the other; • prove the different rules of differentiation using the increment method; • find the derivative of an algebraic function using the basic rules of differentiation; and • extend these basic rules to other “complex” algebraic functions.

Derivative of a Function The process of finding the derivative of a function is called differentiation and the branch of calculus that deals with this process is called differential calculus. Differentiation is an important mathematical tool in physics, mechanics, economics and many other disciplines that involve change and motion.

Consider a point on the curve that is distinct from and compute the slope of the secant line through P and Q.

If we let x 2 approach x 1, then the point Q will move along the curve and approach point P. As point Q approaches P, the value of Δx approaches zero and the secant line through P and Q approaches a limiting position, then we will consider that position to be the position of the tangent line at P.

y tangent line secant line x

Thus, we make the following definition. DEFINITION: Suppose that x 1 is in the domain of the function f, the tangent line to the curve y=f(x) at the point P(x 1, f(x 1)) is the line with equation, where the limit exists, and of tangency. provided is the point

DEFINITION The derivative of y = f(x) at point P on the curve is equal to the slope of the tangent line at P, thus the derivative of the function f given by y= f(x) with respect to x at any x in its domain is defined as: provided the limit exists.

Other notations for the derivative of a function are: Note: To find the slope of the tangent line to the curve at point P means that we are to find the value of the derivative at that point P. There are two ways of finding the derivative of a function: 1. By using the increment method or the four-step rule 2. By using the differentiation formulas

THE INCREMENT METHOD OR THE FOUR-STEP RULE One method of determining the derivative of a function is the increment method or more commonly known as the four-step rule. The procedure is as follows : STEP 1: Substitute x + Δx for x and y + Δy for y in y = f(x) STEP 2: Subtract y = f(x) from the result of step 1 to obtain Δy in terms of x and Δx

STEP 3: Divide both sides of step 2 by Δx. STEP 4: Find the limit of step 3 as Δx approaches 0.

EXAMPLE