ACJC 2025 Differentiation and Applications Lecture Notes

1 5 DIFFERENTIATION SYLLABUS ▪ Graphical interpretation of (i) f ( ) 0, f ( ) 0 and f ( ) 0x x x   =  (ii) f ( ) 0 and f ( ) 0xx    ▪ Relating the graph of f ( )yx = to the graph of f ( )yx= ▪ Differentiation of simple functions defined implicitly or parametrically ▪ Determining the nature of the stationary points (local maximum and minimum points and points of inflexion) analytically, in simple cases, using the first derivative test or the second derivative test ▪ Locating maximum and minimum points using a graphing calculator or a graphing software ▪ Finding the approximate value of a derivative at a given point using a graphing calculator or a graphing software ▪ Problems involving tangents and normals to curves, including cases where the curve is defined implicitly or parametrically ▪ Local maxima and minima problems ▪ Connected rates of change problems

ACJC 2025/26 H2 Mathematics 2 CONTENTS 0 Recap of ‘O’ Level Differentiation .................................................. 3 1 Limits ................................................................................................ 4 2 First Principles .................................................................................. 7 3 Some Rules of Differentiation .......................................................... 9 4 Higher Order Derivatives, d d n n y x ..................................................... 11 5 Derivatives of Standard Functions ................................................. 12 5.1 Derivatives of Trigonometric Functions ............................... 12 5.2 Derivatives of Inverse Trigonometric Functions .................. 13 5.3 Derivatives of Logarithmic Functions .................................. 15 5.4 Derivatives of Exponential Functions ................................... 16 6 Derivatives of Functions Defined Implicitly .................................. 17 7 Derivatives of Functions Defined Parametrically .......................... 22 7.1 Application to Motion of an Object ...................................... 24 8 Differentiation and Graphs ............................................................. 26 8.1 Increasing and Decreasing Functions ................................... 26 8.2 Concavity .............................................................................. 28 8.3 Stationary Points ................................................................... 28 8.4 Points of Inflexion ................................................................. 32 9 Relating the Graph of f ( )yx = to the Graph of f ( )yx= ............ 34 10 Applications of Differentiation....................................................... 41 10.1 Gradients, Tangents and Normals ......................................... 41 10.2 Maxima and Minima ............................................................. 48 10.3 Rates of Change .................................................................... 55 Annex A: Practice Questions on Differentiation ..................................... 60

5 Differentiation 3 LECTURE 1 Lesson Outline • Recap on simple differentiation • Definition of limits and derivative of simple functions from first principles • Revision on some rules of differentiation 0 RECAP OF ‘O’ LEVEL DIFFERENTIATION Type of function Derivative Constants d ( ) 0d cx = , where c is a constant Polynomials ( ) 1d d nnx nxx −= , where n is a real constant Logarithmic ( )d1 lnd xxx = Type equation here. Exponential ( )d eed xx x = Trigonometric ( ) ( ) ( ) 2 d sin cosd d cos sind d tan secd xxx xxx xxx = =− = Self-Practice Exercise 1 Differentiate the following functions with respect to x: (a) πsin 2 2x + (b) 3(2 5)x + (c) 2 e2x + (d) ( )5 ln 3 4xx−+ Ans: (a) π2cos 2 2x + (b) 23(2 5) (2)x + (c) 2 2e xx (d) 35 34x− +

ACJC 2025/26 H2 Mathematics 4 1 LIMITS Examples 1. A regular polygon of n sides is inscribed in a circle. As n increases infinitely (i.e. as n → ), what is the limit of the polygon? As n → , the polygon approaches a circle. Let nA be the area of the inscribed polygon with n sides. As n increases, it appears that nA becomes closer and closer to the area of the circle. We say that the area of the circle, A is the limit of the areas of the inscribed polygons, and we write lim nn AA → = . 2. To find the area of region in Figure 1, we approximate the desired area A by considering areas of n rectangles as shown in Figure 2. As n increases, the width of the rectangles decreases. We calculate A as the limit of these sums of areas of rectangles (this will be discussed in Chapter 10). Definition Given a function f ( )x , if f ( )x approaches a fixed number L as x approaches a, then limf ( ) xa xL → = . y x O A 1 y x O 1 y x O 1 Figure 1 Figure 2 Figure 3 y x O a L

5 Differentiation 5 Example 1 Find the following limits without using the graphing calculator. (a) 2 2 lim ( 5) y y → + (b) 2 20 43lim x xx xx→  +  − (c) 2 2 41lim x x xx→  +  − Solution (a) 2 2 lim ( 5) y y → + 225 9 =+ = (b) 2 20 43lim x xx xx→  +  − 0 0 (4 3)lim ( 1) 43lim 2 x x xx xx x x → →  +=  − +=  − 03 01 3 += − =− (c) 2 2 41lim x x xx→  +  − 2 2 2 2 2 41 1 10 lim 4lim 1 x x xxx x x x x + −→ →  =  +=  − 40 10 4 += − = ■

ACJC 2025/26 H2 Mathematics 6 Example 2 Find the following limits using the graphing calculator. (a) 0 sinlim x x x→   (b) 0 e1lim x x x→  −  (See Tutorial 10 Q1) Solution 1) Press [Y=] and enter the function sin (x) / x. 2) For table setup, press [2nd] [WINDOW]. Change TblStart to 0.01 and Δ Tbl to −0.001. 3) For table, press [2nd] [GRAPH]. From the table obtained, we observe that when the value of x approaches 0, the function approaches 1. Hence, 0 sinlim 1 x x x→  = . Note The graph of sin xy x= has the following shape, but the graph is not defined at 0x = . (a) Using GC, 0 sinlim x x x→  = 1. (b) 0 e1lim x x x→  −  = 1. ■

5 Differentiation 7 2 FIRST PRINCIPLES The gradient of a curve at any point on the curve is defined as the gradient of the tangent at that point. Let ( , f ( ))P x x be any point on the curve f ( )yx= . Let Q be another point on the curve near P, i.e., ( ),f ( )Q x x x x++ where x is a small increment to the value of x. Gradient of the secant line f ( ) f ( )y x x xPQ xx   +−== . As QP→ , i.e. 0x → The secant line PQ → The tangent at P Gradient of the secant line PQ → Gradient of the tangent at P Gradient of the curve at P = gradient of the tangent at P, 00 d f ( ) f ( )lim limd xx y y x x x x x x  →→ +−== Notation 1. We write 0 lim x y x  → as ( )dd or f ( ) or f ( )dd y xxxx  . 2. The process of obtaining d d y x is called differentiation. 3. d d y x is read as ‘first derivative of y with respect to x’. It is the rate of change of y with respect to x. P Q y x This process is called differentiating from First Principles. GeoGebra demonstration https://www.geogebr a.org/m/swsuu5en

ACJC 2025/26 H2 Mathematics 8 4. d dx is an operator and is not a fraction or ratio and dd ()dd y yxx= . 5. To denote the value of the derivative at a specific point xa= , we may write d d xa y x = or f ( )a . Example 3 By considering the derivative as a limit, find the derivative of (i) 2x (ii) 1 x (iii) sin x (iv) ex Solution (i) Let 2f ( )xx = 0 f ( ) f ( )f ( ) lim x x x xx x  → +− = 22 0 2 2 2 0 0 0 ()lim 2 ( )lim (2 )lim lim (2 ) 2 x x x x x x x x x x x x x x x x x x xx x            → → → → +−= + + −= += =+ = (ii) d1 dxx   11 0 0 0 0 2 lim ()lim () lim () 1lim () 1 x x x x x x x x x x x x