Victoria School AM #12 Differentiation 2024

Secondary 4 Additional Mathematics Differentiation 1 Name: Class Reg No. Date: 4 Chapter 12: Differentiation Reference Book: Additional Maths 360 Volume B, Marshall Cavendish You will learn how to, • Relate the derivative of a function to the gradient of tangent. • Differentiate functions using the basic rules of differentiation, ➔ Constant multiple rule: ( ) ( ) dd dd ff xx a x a x =       , where a is a constant ➔ Power rule: d dx x n = nx n – 1 , n is a rational number ➔ Sum and difference rule: ( ) ( ) ( ) ( ) d d d d d d f g f g x x x x x x x =            ➔ Chain rule: d d d d d d y y u x u x=  ( ) 1dd dd nn yu nuxx −= to differentiate functions of the form y = u n where u = f(x) ➔ Product rule: ( )d d d d d d vuuv u vx x x =+ ➔ Quotient rule: 2 dd d dd d uvvuu xx x v v − = Introduction The word "calculus" comes from Latin (calculus) and means a small stone used for counting. Calculus is the mathematical study of change and has two major branches; Differential calculus, which concerns itself with the rate of change and the gradient of curves, as well as integral calculus, which is regarding the accumulation of quantities and area under curves. The First Derivative of y The first derivative of y is denoted by f ( )x or , pronounced as “dee y dee x”. It is the gradient function of y, that is, it is the function that provides the value of the instantaneous gradient of y with respect to x at any required point. ➔ Hence, the gradient (at a particular point) of a curve is defined as the derivative of a function or gradient function. It is the result of differentiating y with respect to x. The process of finding d d y x is called differentiation. Teachers’

Secondary 4 Additional Mathematics Differentiation 2 Notation The notation d d y x , is also known as the first derivative of y with respect to the variable x. If f is a function of x, i.e. f(x), the first derivative of f with respect to x can be denoted by → If y = f(x), then d f ( )d y xx = . Sometimes, you may be asked to find ( )d d nxx . This represents the first derivative of xn with respect to x. ( )d dx is the differential operator. You simply need to differentiate the expression in the brackets with respect to the variable x. Similarly, ( )d dt is the differential operator with respect to the variable t. t is commonly used to represent time. The Power Rule *We “bring down” the power of the variable, followed by reducing the power by one. The Constant Multiple Rule The Constant Rule If nyx= , where n is a rational constant, then 1d d ny nxx −= If ( )f x is a function and a is a constant, then ( ) ( ) dd dd ff xx a x a x =       Given a general function ,yk= where k is a constant, we can use Constant Multiple Rule and the Power Rule to show that d 0.d y x = ( ) ( ) ( ) 0 0 01 dd ()dd d d d d 0 0 y kxx kxx kx x kx − = = = = = Constant Multiple Rule Power Rule

Secondary 4 Additional Mathematics Differentiation 3 Class Practice 1: Differentiate the following functions with respect to x. y d d y x y d d y x a x2 (2)x1 = 2x b x5 (5)x4 = 5x4 c 3x2 3(2)x1 = 6x d 5x4 5(4)x3 = 20 x3 e 10x 910πx f 7 212x − − 9 242x − g 6 7x 67 6 x h 7 417x 3 4119 4 x i 923x− 10207x−− j 3 27 x 3 4 9 x − The Sum Rule and Difference Rule Example 1 Differentiate 2 4xx+ with respect to x. 2d d 4 2 4 x x x x + = + Example 2 Differentiate 32 4 5xx− + − with respect to x. ( ) 3 1 3 2 1 22 2 d d d d 2 4 5 2 4 5 26 2 OR 6 x x xx xx x x x x − − + − = − + − =− + − + If f ( ) g( )y x x= , ( ) ( ) ( ) ( ) d d d d d d f g f g x x x x x x x =           

Secondary 4 Additional Mathematics Differentiation 4 Class Practice 2: 1 Differentiate the following functions with respect to x. (a) 2 2 4 2 35 xx − (b) 893 20 11 xx − 2 2 4 23 d π2(a) d 3 5 2π8 35 xx x xx  − =− 89 78 78 d3 π(b) d 20 11 24 9 π 20 11 69 π 5 11 xx x xx xx  − =− =− 2 Differentiate the following functions with respect to x. (a) 3 61 3xx−+ (b) 3 2 3xx+− ( ) 3 31 42 42 d 6 1(a) 3 d d 63d 18 18 1 x x x xxx xx xx −− −−  −+ = − + =− + =− + ( ) 1 2 1 2 d(b) 3 2 3d 132 2 3 13 xxx x x x − − +− =+   =+ =+ 3 Differentiate 262 , 0,2 xx xx −+  with respect to x. 2 1 12 3 22 23 d 6 2 d2 d1 3d2 13 4 113 4 xx xx x x xx xx xx − − − −  −+ = − +  = + − = + −

Secondary 4 Additional Mathematics Differentiation 5 4 Differentiate the following functions with respect to x. (a) ( )( )g( ) 1 1x x x= + − (b) ( ) 232t x x=− ( )( ) ( )   22 d(a) g ( ) 1 1 d d 1d d 1d 1 x x x x xx xx  = + − =−  =− =− ( ) ( ) 2 5 2 2 3 2 3 dd(b) 3 2dd dd 63dd 1512 2 1512 2 t x xxx t xxxx xx xx =−  =−   =− =− 5 Calculate the gradient of each curve at the given point (a) ( ) 25 4 2, at 1, 3y x x= − + (b) ( )( )1 2 3 , at 2xxyx x −+= =− (a) 25 4 2y x x= − + ( ) 2dd 5 4 2dd 10 4 y xxxx x = − + =− ( ) ( )dAt 1, 3 , 10 1 4d 6 y x =− = ( )( ) 2 1 2 3(b) 23 321 xxy x xx x x x −+= +−= = + − 2 d d 3 21dd 32 y xx x x x = + − =+ At 2,x=− ( ) 2 d3 2d 2 y x =+ − 32 4=

Secondary 4 Additional Mathematics Differentiation 6 6 Given that the equation of a curve is 10 ,yx x=− find the coordinates of the points on the curve at which the gradient is 7 .2− 10yx x=− ( ) 1 2 d d 10 dd d 10d 10 1 y xx x x xxx x − − =−  =− =− − When the gradient is 7 ,2− d7 d2 y x =− 2 2 2 710 1 2 510 2 4 2 or 2 x x x x − − − − =− = = =− When 102, 2 3 2xy= = − = When ( )102, 2 3 2xy=− = − − =−−  the coordinates of the points are (2, 3) and (−2, −3).

Secondary 4 Additional Mathematics Differentiation 7 7 A curve has the equation 3y x px q= + + where p and q are constants. The gradient of the curve at the point (3, 16) is 20. (i) Find the value of p and of q. (ii) Find the coordinates of the other point on the curve where the gradient is 20. (i) 3y x px q= + + Since the point (3, 16) lies on the curve, ( ) ( ) ( ) 3 16 3 3 3 11 1 pq pq = + + + =− − 2d 3d y xpx =+ Substituting d3 and 20,d yx x== ( ) 2 20 3 3 7 p p =+ =− Substituting ( )7 into 1 ,p=− ( )3 7 11 10 q q − + =− = (ii) dGiven: 20d y x = 2 2 2 3 7 20 3 27 9 3 or 3 x x x x −= = = =− When ( ) ( ) 3 3, 3 7 3 10xy=− = − − − + 4= The coordinates of the other point where the gradient is 20 are ( )3, 4 .−

Secondary 4 Additional Mathematics Differentiation 8 The Chain Rule How would you differentiate ( ) 2232x + w.r.t. x? How would you differentiate ( ) 5232x + w.r.t. x? To expand first then differentiate is rather tedious. Is there a shorter way? To obtain the above solution quickly, we apply the Chain Rule. To differentiate ( ) 5232yx=+ , let 23 2.ux=+ 5yu= Note, y is a function of u and u is a function of x. 4dd 5,dd yu uux== 6x ( ) ( ) 4 42 6 d d d d d d 5 30 3 2 x y y u x u x u xx = = =+ If f ( )yu= and g( ),ux= dd and dd yu ux exist, then the derivative of the function ( )fgyx=  exists and is given by, d d d d d d y y u x u x= Note: The Chain Rule looks as if the du’s are cancelled, when in fact, d d u x is a single notation that cannot be taken as a fraction.

Secondary 4 Additional Mathematics Differentiation 9 Example 3 Differentiate 354x + with respect to x. ( ) ( ) 1 23 2 3 1 3 2 2 3 15 54 2 d 54d d 54d 15 2 5 4 xx xx xx x x − +  + =+  = = + Example 4 Differentiate 8 2 2x x  + with respect to x. 7 2 2 8 2 2282 d2 d xx xx xxx +−  +    =   