ECON 213 ELEMENTS OF MATHS FOR ECONOMISTS Session 7 – Differential Calculus- Part One Lecturer: Dr. Monica Lambon-Quayefio Contact Information: mplambon-quayefio@ug.edu.gh

Session Overview Overview Many disciplines including economics are usually interested in how quickly quantities change over time. In order to predict or estimate the future demand of a commodity or predict the growth in population etc, we need information about the rates of change. This session introduces students to the concept of derivatives and presents some of the important rules for calculating it. Objectives Be able to define the derivative of a function and determine the derivatives of various function types Know the simple rules for differentiation: sum rule, the product rule, the quotient rule and the chain rule.

Session Outline The key topics to be covered in the session are as follows: Definition of Derivatives and derivatives of various function types Rules of Differentiation

Reading List Sydsaeter, K. and P. Hammond, Essential Mathematics for Economic Analysis, 2nd Edition, Prentice Hall, 2006- Chapter 6 Dowling, E. T., “Introduction to Mathematical Economics”, 3rdEdition, Shaum’s Outline Series, McGraw-Hill Inc., 2001.- Chapter 3 Chiang, A. C., “Fundamental Methods of Mathematical Economics”, McGraw Hill Book Co., New York, 1984.- Chapter 7

Derivatives: definition and derivatives of various functions Topic One Derivatives: definition and derivatives of various functions

Derivatives: Definition From the equation of a line y=mx +c , m denotes the slope of the line. If m is large and positive, then the line rises steeply from left to right and if it is m is large and negative, the lines falls steeply. For any arbitrary function f, what is the steepness of its graph? The natural answer is to define the steepness of a curve at a particular point as the slope of the tangent to the curve at that point. In other words, the steepness of a curve/ function at a particular point is defined as the slope of the line that touches the curve at that point. Assume that P is a particular point on a curve. Then the steepness of the curve at point P is the slope of the tangent at point p which is called the derivative of the curve/function at that point

Definition Contd. Formally, the derivative of any function at a particular point a, denoted by denoted by f ′(a), is given by the relationship if the limit exists. Various notations used are as follows:

3.1 Derivative of a Function “the derivative of f with respect to x” “y prime” “the derivative of y with respect to x” “the derivative of f of x”

Differentiability. To be differentiable, a function must be continuous and smooth. Derivatives will not exist at the following: cusp corner vertical tangent discontinuity

Differentiability: Intermediate value theorem If a and b are any two points in an interval on which f is differentiable, then takes on every value between and . Between a and b, must take on every value between and .

Rules of differentiation Topic Two Rules of differentiation

Differentiation: Function Types The constant Function: Let f(x)= k where k is a constant 𝑓 ′ (x)= (𝑘) ′ = 0 Examples: (8) ′ = 0 (−2) ′ =0 (0.5) ′ = 0 The identity function: Let f(x)= x then the 𝑓 ′ (x)= (𝑥) ′ = 1

Differentiation: Function Types A function of the form 𝑥 𝑛 Let f(x)= 𝑥 𝑛 , a function of x, and n a real constant. We have 𝑓 ′ (x)= ( 𝑥 𝑛 ) ′ = n 𝑥 𝑛−1 Examples: 𝑥 5 = ( 𝑥 5 ) ′ = 5 𝑥 5−1 = 5 𝑥 4 𝑥 −2 = ( 𝑥 −2 ) ′ = -2 𝑥 −2−1 = -2 𝑥 −3 𝑥 1/2 = ( 𝑥 1/2 ) ′ = 1/2 𝑥 1/2−1 = 1/2 𝑥 −1/2

Differentiation: Function Types Exponential Function of the form 𝑎 𝑥 : Let f(x) = 𝑎 𝑥 , where a>0 and x is a variable, we have 𝑓 ′ (x)= ( 𝑎 𝑥 ) ′ = 𝑎 𝑥 ln (a) Examples: ( 4 𝑥 ) ′ = 4 𝑥 ln (4) ( 1 2 𝑥 ) ′ = 1 2 𝑥 ln ( 1 2 ) The function 𝑒 𝑥 : Let the function f(x) = 𝑒 𝑥 . Then 𝑓 ′ (x)= ( 𝑒 𝑥 ) ′ = 𝑒 𝑥

Differentiation: Function Types The Logarithmic function ln(x): Given the logarithmic function f(x) =ln(x), we have 𝑓 ′ (x)= (ln⁡(𝑥) ′ = ln ( 1 𝑥 ) Examples: ( 4 𝑥 ) ′ = 4 𝑥 ln (4) ( 1 2 𝑥 ) ′ = 1 2 𝑥 ln ( 1 2 ) The function 𝑒 𝑥 : Let the function f(x) = 𝑒 𝑥 . Then 𝑓 ′ (x)= ( 𝑒 𝑥 ) ′ = 𝑒 𝑥

Topic Three Rules of derivatives

Rules of Differentiation Constant Multiples: Let k be a constant and f(x) be any given function. Then (k (f( 𝑥)) ′ = k f( 𝑥) ′ Examples: ( 4𝑥 2 )’= 8x (-5 𝑒 𝑥 )’= -5 𝑒 𝑥 (13 ln(x))’= 13( 1 𝑥 )= 13 𝑥 2. Addition and Subtraction Function Let f(x) and g(x) be two functions. Then ((𝑓(𝑥)±𝑔(𝑥))= (𝑓( 𝑥) ′ ±𝑔( 𝑥) ′

Rules of Differentiation Examples: ( 𝑒 𝑥 + 𝑥 4 ) ′ = ( 𝑒 𝑥 ) ′ + ( 𝑥 4 ) ′ = 𝑒 𝑥 + 4 𝑥 3 (ln x - 1 𝑥 3 + 4)= ( ln⁡(𝑥)) ′ - ( 𝑥 −3 ) ′ + ( 4) ′ = 1 𝑥 + 3𝑥 −4 +0 2. Product Rule Let f(x) and g(x) be two functions. Then the derivative of the product ((f 𝑥 𝑔(𝑥)) ′ = 𝑓( 𝑥) ′ g x +f x g( 𝑥) ′ Example: ( 𝑥 2 𝑒 𝑥 ) ′ = ( 𝑥 2 ) ′ 𝑒 𝑥 + 𝑥 2 ( 𝑒 𝑥 ) ′ = 2x 𝑒 𝑥 + 𝑥 2 𝑒 𝑥

Rules of Differentiation 2. Quotient Rule Let f(x) and g(x) be two functions. Then the derivative of the quotient is given as: ((f 𝑥 /𝑔(𝑥)) ′ = 𝑓( 𝑥) ′ g x −f x g( 𝑥) ′ (g( 𝑥) 2 Example ( 𝑥 2 /𝑒 𝑥 ) ′ = ( 𝑥 2 ) ′ 𝑒 𝑥 − 𝑥 2 ( 𝑒 𝑥 ) ′ = 2x 𝑒 𝑥 − 𝑥 2 𝑒 𝑥 ( 𝑒 𝑥 ) 2 (𝑒 𝑥 ) 2

Rules of Differentiation Derivative of a composite function: A composite function is a function with the form f(g(x). A composite function is in fact a function that contains another function. If you have a function that can be broken down into many parts, where each part is in itself a function and where these parts are not linked by addition, subtraction, product or division, you usually have a composite function For example, the function f(x)= 𝑒 𝑥 4 is a composite function. We can rewrite this function as f(g(x) where g(x)= 𝑥 4 . Note however that f(x)= 𝑥 4 𝑒 𝑥 is not a composite function since it is only the product of two functions.

Examples of Composite Functions F(x) = ln ( 𝑥 3 +2 𝑥 2 -x+6) This function can be rewritten as f(g(x) where g(x)= ( 𝑥 3 +2 𝑥 2 -x+6) F(x) = 𝑒 3𝑥−2 This function can be rewritten as f(g(x) where g(x)= 3x-2 F(x)= ( ln 𝑥 −7𝑥+ 𝑒 𝑥 ) 2 g(x)= ln(x)-7x+ 𝑒 𝑥

The Chain Rule of Composite Functions Let f and g be two functions. Then the derivative of the two composite functions is given as (f( 𝑔(𝑥))) ′ = 𝑓 ′ (g(x)) 𝑔 ′ (x) or (f( 𝑢)) ′ = 𝑓 ′ (u) 𝑢 ′ where u = g(x) The chain rule states that when we derive a composite function, we must first derive the external function (the one which contains all others) by keeping the internal function as is and then multiplying it with the derivative of the internal function

Examples [ ln⁡( 𝑥 2 +4𝑥−1] ′ = 1 𝑥 2 +4𝑥−1 ( 𝑥 2 +4𝑥−1) ′ = 2𝑥+4 𝑥 2 +4𝑥−1 ( 𝑒 3𝑥 2−6 ) ′ = 𝑒 3𝑥 2−6 ( 3𝑥 2 -6)’ = 𝑒 3𝑥 2−6 (6x-6 )

Session Problem Set 1. Find the first order derivatives of the following functions: y= 𝑒 𝑥 + 𝑥 2 y= 5 𝑒 𝑥 -3 𝑥 2 +8 y= 𝑥+ 𝑥 2 𝑒 𝑥 +1 2. Using the chain rule, differentiate the following y= ( 𝑥 3 + 𝑥 2 ) 10 y= ( 𝑥−1 𝑥+3 ) 1 3 y= ( 𝑥 2 +1)