2024 JPJC Chapter 6 Techniques of Differentiation

Jurong Pioneer Junior College H2 Mathematics (9758) JC1-2023 Chapter 6 Techniques of Differentiation (Students’ Version)/ Pg 1 Chapter 6: Techniques of Differentiation Content Outline  Differentiation of simple functions  Differentiation of simple functions defined implicitly or parametrically  Finding the approximate value of a derivative at a given point using a graphing calculator Derivatives listed in MF26 f( )x f '( )x 1sin x 2 1 1 x 1cos x 2 1 1 x   1tan x 2 1 1 x cosec x  cosec x cot x sec x sec tanx x References  http://www.h2maths.site [Demonstration on various differentiation techniques thro ugh keying in different expressions.]  http://www.calculus-help.com/tutorials/ [Animated demonstration on various differentiation techniques.]  http://www.mathbits.com/MathBits/TISection/Openpage.htm [Using TI Graphing Calculator in differentiation.]  AS: Use of Maths – Calculus (Publisher: Nelson Thornes)  Calculus DeMystified – by Steven G. Krantz  Calculus – The Easy Way – by Douglas Downing Prerequisite Secondary school knowledge of calculus (d ifferentiation), algebra and coordinate geometry.

Jurong Pioneer Junior College H2 Mathematics (9758) JC1-2023 Chapter 6 Techniques of Differentiation (Students’ Version)/ Pg 2 Introduction: What is Calculus? Calculus is the mathematics of motion and change, which is why calculus is a prerequisite fo r many courses. Whenever we move from the static to the dynamic, we would consider using calculus. In the 17 th century, c alculus was developed and researched in attempt to answer some fundamental questions about the world and the way things work. These inv estigations led to two fundamental concepts of calculus – derivative and integral. The breakthrough in the development of these concepts was the formulation of a mathematical tool called a limit. 1. Definitions (i) Gradient The gradient of a function f ( )x defines the direction of the graph of f( x) and shows how the function f(x) changes with x. (a) Linear function: y mx c  The gradient of a linear function y mx c  is the constant m where m is the tangent of the angle that the line makes with the positive direction of x-axis. Gradient of the line l = m = 2 1 2 1 y y x x   = tan Note: 0 in the above case since 0 2m    . For the case below, since , 02 m     .

Jurong Pioneer Junior College H2 Mathematics (9758) JC1-2023 Chapter 6 Techniques of Differentiation (Students’ Version)/ Pg 3 (b) Non-linear function: The gradient of a curve at any point is defined to be the gradient of the tangent drawn at that point. Thus, the gradient of a curve is not a constant but has different values at different points on the curve. (ii) Differentiation The process of determining the rate of change of a function with respect to one of its variables, e.g. the rate of change of y with respect to x, is known as differentiation. The general expression for the gradient is called the derivative or the gradient function and is denoted by the symbol d d y x . The derivative of a function f ( )x with respect to x is denoted by f '( )x . 2. Differentiation of Basic Functions (from ‘O’ Level Mathematics) Differentiation with respect to x Results 1  d d naxx (note that a and n are constants) 1nnax  (i)  d d axx (note that a is a constant) a (ii)  d d ax (note that a is a constant) 0 2  d sind xx cos x 3  d cosd xx sin x 4  d tand xx 2sec x 5  d d ex x e x 6  d lnd xx 1 x Note: You are required to remember the above differentiation results. x y

Jurong Pioneer Junior College H2 Mathematics (9758) JC1-2023 Chapter 6 Techniques of Differentiation (Students’ Version)/ Pg 4 3. Basic Rules of Differentiation 1 Sum/Difference Rule  d d d d d d u vu vx x x   (Note: u and v are functions of x.) 2 Product Rule d d d( )d d d v uuv u vx x x   (Note: u and v are functions of x.) 3 Quotient Rule 2 d d d d d d u vv uu x x x v v      (Note: u and v are functions of x.) 4. Chain Rule Chain rule is a process that allows us to differentiate composite functions e.g.   532 1x   , sin 3x ,  ln ln x etc. To find the derivative of a composite function, we use: How chain rule works: Find 4d (2 1)d xx  . Let 2 1 u x  and 4 y u Then d 2d u x  , 3d 4d y uu          3 3 3 d d d d d d 4 2 4 2 1 2 8 2 1 y y u x u x u x x        Students’ note: Composite functions can be seen as combination of basic functions. (i.e. basic function within another basic function) d d d d d d y y u x u x 

Jurong Pioneer Junior College H2 Mathematics (9758) JC1-2023 Chapter 6 Techniques of Differentiation (Students’ Version)/ Pg 5 Example 1 Find   2d ed x x . Solution:   2 2 2 2d de e 2 ed d x x x x xx x       Example 2 Find  d sin 3d xx . Solution:      d dsin 3 cos3 3d d 3cos 3 x x xx x x       Example 3 Differentiate the following functions with respect to x: (a) 1 211 x     ; (b) 3cos 2 x . Solution: (a) 1 1 2 2 2 d 1 1 1 11 1d 2x x x x                     (b)     33d dcos 2 cos 2d d x xx x       2 2 3 cos 2 sin 2 2 = 6sin 2 cos 2 x x x x   

Jurong Pioneer Junior College H2 Mathematics (9758) JC1-2023 Chapter 6 Techniques of Differentiation (Students’ Version)/ Pg 6 Example 4 Differentiate the following exponential and logarithmic functions with respect to x: (a) e x ; (b) ln(sin )x ; (c) ln( )px ; (d) 3log x ; (e)   2lg 1x  . Solution: (a)     1 2d 1 1e e ed 2 2 x x x xx x         (b)  d 1ln(sin ) (cos ) cotd sin x x xx x   (c)    d 1 1ln( ) d px px px x   Alternatively    d d 1 1ln( ) ln ln 0d d px p xx x x x      (d)  3 d d lnlogd d ln 3 xxx x      1 d 1 1 1 (ln )ln 3 d ln 3 ln 3xx x x                   (e)   2 2 2d d ln( 1) 1 dlg 1 ln( 1)d d ln10 ln10 d xx xx x x                  2 2 1 1 2 2 ln10 1 ln10 1 xxx x         Recall: Before differentiating a logarithmic function, simplify the function first using the following Laws of Logarithms: Laws of Logarithms: For all 0, 0 and 0, 1m n a a    , (i) log logk a am k m (Power Law) (ii)  log log loga a amn m n  (Product Law) (iii) log log loga a a m m nn       (Quotient Law) Change of Base of Logarithms: For all 0, 0, 1 and b 0, 1m a a b     , loglog log b a b mm a In particular, lnlog ln a mm a Note: “lg” means “ 10log ”. Question: Is there an equivalent differentiation formula for loga ?

Jurong Pioneer Junior College H2 Mathematics (9758) JC1-2023 Chapter 6 Techniques of Differentiation (Students’ Version)/ Pg 7 5. Differentiation of Other Basic Functions (Trigonometric) Differentiation with respect to x Results 1  d secd xx sec tanx x 2  d cosecd xx cosec cotx x 3  d cotd xx 2cosec x 4   1d sind xx  2 1 1 x 5   1d d cos xx  2 1 1 x   6   1d tand xx  2 1 1 x Note: Formulae 1, 2, 4, 5 and 6 are in MF26 (‘A’ level formulae list) while formula 3 is NOT. Hence, you need to remember formula 3. Example 5 Prove that  d cosec cosec cot d x x xx  . Solution:  d cosecd xx = d 1 d sinx x      =    1d sind xx      2 2 sin (cos ) cos si