SAJC Chapter_5_Techniques_of__Differentiation_(Tutor)_2022

SAJC 2022 JC 2 H1 Mathematics Page 1 of 20 Chapter 5 (Pure Mathematics) Techniques of Differentiation Objectives At the end of the chapter, you should be able to: (a) understand the derivative of ( )f x as the gradient of the tangent to the graph of ( )fyx= at a point; (b) use of standard notations ( )f ' x and d d y x ; (c) differentiate nx for any rational n, xe , ln x , together with constant multiples, sums and differences; (d) use the chain rule to differentiate a composition of 2 or 3 functions; (e) find the approximate value of a derivative at a given point using a graphing calculator Content 5.1 Introduction 5.2 Derivatives of Basic Functions 5.3 Rules of Differentiation 5.3.1 Extension of Derivatives of Basic Functions 5.4 Finding Numerical Values of Derivatives for Given Values of x by GC 5.5 Higher Order Derivatives References New Syllabus Additional Mathematics (8th Edition), Shinglee Publishers Pte Ltd. Relevant Resources • http://mathinsite.bmth.ac.uk/applet/diffchords/diffchords.html (Applet to explore graphically the differentiation of a quadratic curve of 2yx= • http://www.slu.edu/classes/maymk/GeoGebra/TangentToDerivatives.html (Applet to explore differentiation of being the slope of the line tangent to a curve)

SAJC 2022 JC 2 H1 Mathematics Page 2 of 20 5.1 Introduction View the following video 1. https://www.youtube.com/watch?time_continue=15&v=EKvHQc3QEow or 2. Scan the QR code on the right. You should see this page: Note: 1. The process of finding the derivative of a function is called differentiation. d d y x means we differentiate the function y with respect to x. 2. If ( )fyx= , we may write ( )dd f f '( )dd y xxxx== ( )f' x is known as the first derivative of ( )f x . 3. For a curve ( )fyx= , d d y x denotes gradient of the tangent to the curve ( )fyx= at the point ( )( ),fxx . 4. d d y x also denotes the rate of change of the function y with respect to x. Alternatively, you can use the QR code:

SAJC 2022 JC 2 H1 Mathematics Page 3 of 20 5.2 Derivatives of Basic Functions Basic Functions y d d y x Algebraic (for any rational n) nx 1nnx − Exponential ex ex Logarithmic ln x 1 x Example 1 (Differentiate algebraic functions) Differentiate the following with respect to x: (a) 6xy = 6 5d 6d yx y xx = = (b) 4yx −= 4 5 5 d 4d 4 yx y xx x − − = =− =− (c) yx= 1 2 1 2 1 2 d1 d2 1 2 1 2 y x x y xx x x − == = = = (d) 6y= 6 d 0d y y x = =

SAJC 2022 JC 2 H1 Mathematics Page 4 of 20 5.3 Rules of Differentiation Suppose ( )f x and ( )g x are functions of x and a and b are constants. Rules Example 1 Constant Multiple of Function     ( ) dd f ( ) f ( )dd f' a x a xxx ax = = 2d 20d xx  = 20(2 ) 40xx= d 5d xex −= 5( ) 5xxee− =− d ln 1 1 1 d 3 3 3 x