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