TMJC H2 Chapter 7 Differentiation Learning Package 2024

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Chapter 7 Differentiation TMJC 2024 Page 1 of 36 H2 Mathematics (9758) Chapter 7 Differentiation Core Concept Notes Success Criteria: Surface Learning Deep Learning Transfer Learning Interpret the derivative d d y x as the gradient of a curve Use a graphing calculator to find the approximate value of a derivative at a given point Differentiate algebraic, trigonometric, logarithmic, exponential and inverse trigo functions Determine the nature of the stationary points (local maximum and minimum points and stationary point of inflexion) analytically, in simple cases, using the first derivative test or second derivative test Locate the maximum and minimum points using a graphing calculator Use chain rule, product rule and quotient rule Interpret graphically   f 0 x  (strictly increasing)   f 0 x  (stationary)   f 0 x  (strictly decreasing)   f 0 x  (concave upwards)   f 0 x  (concave downwards) Use a graphing calculator to investigate the relationship between the graphs of f( )y x and  f 'y x Differentiate functions defined implicitly Find higher derivatives Differentiate functions defined parametrically Relate the graph of  f 'y x to the graph of f( )y x

Chapter 7 Differentiation TMJC 2024 Page 2 of 36 §1 Differentiation In mathematics, differentiation is a subfield of calculus concerned with the study of the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integration, the study of the area under a curve. The primary objects of study in differentiation are the derivative of a function, related notions such as the differential, and their applications. The process of finding a derivative is called differentiation. Geometrically, the derivative at a point is the slope of the tangent line to the graph of the function at that point, provided that the derivative exists and is defined at that point. The modern development of calculus is usually credited to Isaac Newton (1643–1727) and Gottfried Wilhelm Leibniz (1646–1716), who provided independent and unified approaches to differentiation and derivatives. The key insight, however, that earned them this credit, was the fundamental theorem of calculus relating differentiation and integration. Note: For d d d d yyx x   d dx : an operator to differentiate a function with respect to x  d d y x : a function which is the result of differentiating y with respect to x 1.1 Basic Rules of Differentiation Let n and c be constants and u and v be differentiable functions of x i.e. u = g(x) and v = h(x) Basic Rules Example (1)  d 0d cx  (2)    d d d d n ncx c xx x  1ncnx  (3)  d d d d d du v u vx x x   d d d d u v x x          3 3 2 2 d 2 3d d d2 3d d 2 3 0 6 xx xx x x x       Note: Since u = g(x), the derivative d d u x may also be written as g '( )x .

Chapter 7 Differentiation TMJC 2024 Page 3 of 36 1.2 Chain Rule, Product and Quotient Rule of Two Functions Let y, u and v be differentiable functions of x. Examples Chain Rule x u u y x y d d d d d d      6 5 5 5 d d2 7 6(2 7) 2 7d d 6(2 7) (2) 12(2 7) x x xx x x x         Product Rule  d d d d d d v uuv u vx x x                           6 6 6 5 6 5 5 d d d3 7 3 7 7 3d d d 3 6 7 1 7 3 7 18 3 7 21 7 1 x x x x x xx x x x x x x x x x x                          Quotient Rule 2 d d d d d d v x vux uv v u x                               2 2 2 2 2 2 2 2 2 d d2 1 2 1d d d d 2 1 2 1 2 1 2 2 2 1 2 12 2 2 1 2 1 x x x xx x x x x x x x x x x xx x x x                  Note: Do not invent your own product rule or quotient rule!! E.g.       6 6d d d3 7 3 7d d dx x x xx x x    and     2 2 d d d dd 2 1 2 1d xx x x x xx       

Chapter 7 Differentiation TMJC 2024 Page 4 of 36 §2 Differentiation of Logarithmic and Exponential Functions Basic General (Using Chain Rule) Example   d e ed x x x     f ( ) f ( )d e e f 'd x x xx    23d ed x x   23 2 de 3 d x xx  236 e xx   d ln ,d x xa a ax  where 0a   f ( ) f ( )d f ' lnd x xa a x ax    2d 2d x x    2 2 ln 2 2x x  d 1lnd xx x     d 1ln f( ) f 'd f ( ) x xx x    3d ln 1d xx      2 3 1 3 1 x x   2 3 3 1 x x    d 1 1logd ln a xx a x   , where 0, 1.a a        d log f ( )d d ln f ( ) d ln 1 d ln f ( )ln d 1 1 f 'ln f ( ) a xx x x a xa x xa x             2 10 2 2 d log 1d ln 1d d ln10 1 2 ln10 1 xx x x x x                Recall: If a, b and c are positive numbers and 1, 1,a c  then loglog log c a c bb a . Note: Given a function which involves logarithmic functions, try to simplify using the laws of logarithms before carrying out differentiation. Example 1 Differentiate 3 2ln 1 x x      with respect to x. Solution:       3 3 2 2 2 ln ln ln 11 3ln ln 1 x x xx x x              3 2 2 2 d d ln 3ln ln 1d 1 d 3 2 1 x x xx x x x x x              Laws of Natural Logarithm 1) ln ln lnxy x y  2) ln ln lnx x yy   3) ln lnyx y x

Chapter 7 Differentiation TMJC 2024 Page 5 of 36 §3 Differentiation of Trigonometric Functions Basic General (Using Chain Rule) xxx cos)(sind d        d sin f f ' cos fd x x xx  xxx sin)(cosd d         d cos f f ' sin fd x x xx   xxx 2sec)(tand d         2d tan f f ' sec fd x x xx  xxx cosec)(cotd d 2         2d cot f f ' cosec fd x x xx   d (sec ) sec tan (MF27)d x x xx          d secf f ' secf tan f d x x x xx  xxxx cot cosec) cosec(d d  (MF27)          d cosec f f ' cosec f cot fd x x x xx   Note: For trigonometric functions above, x is in radians. Refer to Annex A for the proof of derivative of secx.

Chapter 7 Differentiation TMJC 2024 Page 6 of 36 Example 2 Differentiate (a) 3 sin 2x x with respect to x, (b) 2 2 sin cos  with respect to . Solution: (a)           3 3 2 3 2 2 d sin 2d cos 2 2 sin 2 3 2 cos 2 3 sin 2 2 cos 2 3sin 2 x xx x x x x x x x x x x x x       (b)   2 2 2 d sin cosd 2 cos 2 cos sin         

Chapter 7 Differentiation TMJC 2024 Page 7 of 36 §4 Inverse Trigonometric Functions 4.1 Definition For the three trigonometric functions, their inverse functions are defined for the respective principal range as shown in the table below. Inverse Trigonometric Function Principal Domain Principal Range 1siny x   1,1 i.e. –1 ≤ x ≤ 1 ,2 2      i.e. 2 2 y    1cosy x   1,1  0,  1tany x   ,2 2      Note: 1 1sin sinx x   . Hence )sin 1(d dsind d 1 xxxx  Remember that 1 sin x  is the inverse function and not the same as   1 sin x  , which is the reciprocal function. 4.2 Differentiation of Inverse Trigonometric Functions Basic (MF27) General (Using Chain Rule) 2 1 1 1)(sind d x xx          1 2 d 1sin f ( ) f 'd 1 f ( ) x xx x     2 1 1 1)(cosd d x xx          1 2 d 1cos f ( ) f 'd 1 f ( ) x x