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

DIFFERENTIATION OF ALGEBRAIC FUNCTIONS

DERIVATIVE USING FORMULAS The increment-method (four-step rule) of finding the derivative of a function gives us the basic procedures of differentiation. However these rules are laborious and tedious when the functions to be differentiated are “complex”, that is, functions with large exponents, functions with fractional exponents and other rational functions

Understanding of theorems of differentiation is very important. This is the heart of differential calculus. All of the succeeding topics such as applications of derivatives, differentiation of transcendental functions etc. will be dependent on these theorems. Understanding of these theorems will enable us to calculate derivatives more efficiently and will make calculus easy and enjoyable.

DIFFERENTIATION FORMULAS Derivative of a Constant Theorem: The derivative of a constant function is 0; that is, if c is any real number, then,

DIFFERENTIATION FORMULAS Derivatives of Power Functions Theorem: ( Power Rule) If n is a positive integer, then, In words, to differentiate a power function, decrease the constant exponent by one and multiply the resulting power function by the original exponent.

DIFFERENTIATION FORMULAS Derivative of a Constant Times a Function Theorem: ( Constant Multiple Rule) If f is a differentiable function at x and c is any real number, then cf is also differentiable at x and In words, the derivative of a constant times a function is the constant times the derivative of the function, if this derivative exists.

Proof:

DIFFERENTIATION FORMULAS Derivatives of Sums or Differences Theorem: ( Sum or Difference Rule) If f and g are both differentiable functions at x, then so are f + g and f – g, and or In words, the derivative of a sum or of a difference equals the sum or difference of their derivatives, if these derivatives exist.

Proof:

DIFFERENTIATION FORMULAS Derivative of a Product Theorem: (The Product Rule) If f and g are both differentiable functions at x, then so is the product f ● g, and or In words the derivative of a product of two functions is the first function times the derivative of the second plus the second function times the derivative of the first, if these derivatives exist.

Proof:

DIFFERENTIATION FORMULAS Derivative of a Quotient Theorem: (The Quotient Rule) If f and g are both differentiable functions at x, and if g(x) ≠ 0 then f/g is differentiable at x and or

$In words, the derivative of a quotient of two functions is the fraction whose$

In words, the derivative of a quotient of two functions is the fraction whose numerator is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator and whose denominator is the square of the given denominator Note: The proof is left for the students.

DIFFERENTIATION FORMULAS Derivatives of Composition Theorem: (The Chain Rule) If g is differentiable at x and if f is differentiable at g(x), then the composition f ◦ g is differentiable at x. Moreover, if y=f(g(x)) and u=g(x) then y=f(u) and

DIFFERENTIATION FORMULAS Derivative of a Radical with index equal to 2 If u is a differentiable function of x, then The derivative of a radical whose index is two, is a fraction whose numerator is the derivative of the radicand, and whose denominator is twice the given radical, if the derivative exists.

DIFFERENTIATION FORMULAS Derivative of a Radical with index other than 2 If n is any positive integer and u is a differentiable function of x, then The derivative of the nth root of a given function is the exponent multiplied by the product of u whose power is diminished by one and the derivative of u, if this derivative exists.

EXERCISES I. Find the derivative of the following functions using the four-step rule.

EXERCISES II. Find the derivative of the following functions using the four-step rule.

EXERCISES III. Evaluate the derivative of the following functions using the four-step rule with the indicated value of x.