CHAPTER 2 DIFFERENTIATION 2.1 FIRST ORDER DIFFERENTIATION. What is Differentiation?

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1 BA01 ENGINEERING MATHEMATICS 01 CHAPTER DIFFERENTIATION.1 FIRST ORDER DIFFERENTIATION What is Differentiation? Differentiation is all about finding rates of change of one quantity compared to another. We need differentiation when the rate of change is not constant. Notation for the Derivative IMPORTANT: The derivative (also called differentiation) can be written in several ways. This can cause some confusion when we first learn about differentiation. The following are equivalent ways of writing the first derivative of y = f(): d ' or ( ) or ' f y.1.1 RULES OF DIFFERENTIATION A. Derivative of Power Function y a n So na d n1 Eamples: 1. Find the derivative of y = -7 6 Note: We can do this in one step: We can write: OR y' =

2 BA01 ENGINEERING MATHEMATICS 01 Eample 1 Find the derivative for each of the following function. a) y b) y 5 c) y 4 1 d) y e) y f) y g) y h) y 5 i) 1 y j) y 5 k) y l) y 9

3 BA01 ENGINEERING MATHEMATICS 01 B. Derivative of a Constant Function y a So 0 d Eample Find the derivative for each of the following functions. a) y 1 b) y c) y 40 d) y 1.1. THE DERIVATIVE OF SUMMATION AND SUBSTRACTION If ( ) and ( ) are differentiable functions, the derivative of y f g y f g and f ' g ' d f ' g ' d Eamples: 1. Find the derivative of y = 5-1 y = 5 1 Now, 0

4 BA01 ENGINEERING MATHEMATICS 01 And since we can write: So,. Find the derivative of Now, taking each term in turn: (using ) (using ) (since - = -( 1 ) and so the derivative will be -( 0 ) = -1) So (since ) 1

5 BA01 ENGINEERING MATHEMATICS 01 Eample Find the following derivatives; a) 4 y 6 b) y c) 4 d) p q q s t t e) 1 1 y f) 4 y 9 5 f) y 14 y g) 1

6 BA01 ENGINEERING MATHEMATICS 01 Eercise Find the derivative of the following function; 4 i. f f 6 9 ii. 11 iii. y 8 5 iv. y 1

7 BA01 ENGINEERING MATHEMATICS THE DERIVATIVES OF COMPOSITE FUNCTION Chain Rule If y f u, where, u is a function of, so: du d du d This means we need to 1. Recognise u (always choose the inner-most epression, usually the part inside brackets, or under the square root sign).. Then we need to re-epress y in terms of u.. Then we differentiate y (with respect to u), then we re-epress everything in terms of. 4. The net step is to find du d. 5. Then we multiply du Eample 1: and du d. Differentiate each the following function with respect to. i. y = ( + ) 5 In this case, we let u = + and then y = u 5. We see that: u is a function of and y is a function of u. For the chain rule, we firstly need to find and. So 4

8 BA01 ENGINEERING MATHEMATICS 01 ii. In this case, we let u = 4 and then. Once again, u is a function of and y is a function of u. Using the chain rule, we firstly need to find: and So i. y 4 ii. 1 y 4 9 iii. y 4 5

9 BA01 ENGINEERING MATHEMATICS 01 The Etended Power Rule An etension of the chain rule is the Power Rule for differentiating. We are finding the derivative of u n (a power of a function): n y a b k k a b d d a b n1 n k 1 kan a b d n k 1 n Eample: 1. In the case of we have a power of a function. If we let u = - 1 then y = u 4. So now y is written as a power of u; and u is a function of [ u = f() ]. To find the derivative of such an epression, we can use our new rule: where u = - 1 and n = 4. So 6

10 BA01 ENGINEERING MATHEMATICS 01 We could, of course, use the chain rule, as before: d du * du d a) 5 y 4 b) y 4 c) y 4 d) 8 y 6 7 e) y 1 5 f) y 7 1 g) y 5 1 h) y i) 7 y 4 7

11 BA01 ENGINEERING MATHEMATICS DERIVATIVE OF A PRODUCT FUNCTION If u and v are two functions of, then the derivative of the product uv is given by... In words, this can be remembered as: "The derivative of a product of two functions is the first times the derivative of the second, plus the second times the derivative of the first." Eample: If we have a product like y = ( + 6)( + 5 ) we can find the derivative without multiplying out the epression on the right. We use the substitutions u = + 6 and v = + 5. We can then use the PRODUCT RULE: We first find: and Then we can write: 8

12 BA01 ENGINEERING MATHEMATICS 01 Eercise: a) y 4 5 b) y 1 1 c) y 1 4 e) y 8 1 d) y f) y DERIVATIVE OF A QUOTIENT FUNCTION (A quotient is just a fraction.) If u and v are two functions of, then the derivative of the quotient u/v is given by... In words, this can be remembered as: 9

13 BA01 ENGINEERING MATHEMATICS 01 "The derivative of a quotient equals bottom times derivative of top minus top times derivative of the bottom, divided by bottom squared." Eample: 1. We wish to find the derivative of the epression: Solution: We recognise that it is in the form:. We can use the substitutions: u = and v = 4 Using the quotient rule, we first need to find: And Then 40

14 BA01 ENGINEERING MATHEMATICS 01. Find if. Solution We can use the substitutions: u = 4 and v = + Using the quotient rule, we first need: Then and 41

15 BA01 ENGINEERING MATHEMATICS 01 a) y 5 4 b) y 8 c) y d) y t e) g s s 5 s f) g 4 4

16 BA01 ENGINEERING MATHEMATICS 01 Challenge Find the derivative of 4

17 BA01 ENGINEERING MATHEMATICS DERIVATIVE OF LOGARITHMIC FUNCTION If, y ln 1 d y ln u 1 d u d a ln a b d a b Eample: Differentiate each of the following functions; i. y ln 1 d d d 1 1 iv. yln 1 ii. y ln 4 iii. yln 5 44

18 BA01 ENGINEERING MATHEMATICS 01 Eercise: 5 1. y ln 4. yln 7. y ln 4 4. y ln 1 5. y ln y ln 1 45

19 BA01 ENGINEERING MATHEMATICS DERIVATIVE OF EXPONENTIAL FUNCTION If, So, y e e d Eample: Differentiate each of the following functions; i. y e 4 e d e 4 4 4e d 4 d 4 4 ii. y e e d e 6e d d 6 iii. y e d e d d e 46

20 BA01 ENGINEERING MATHEMATICS 01 Eercise: y 1 e 1 ii. y e iii. i. 5 y e iv. y e v. vi. y 1e y e ln 47

21 BA01 ENGINEERING MATHEMATICS DERIVATIVE OF TRIGONOMETRY FUNCTIONS If, y sin cos d y cos sin d y tan sec d Eample: Differentiate each of the following with respect to ; i. y sin v. y sin d cos d d cos 1 cos ii. y cos d sin d d sin 6sin iii. y tan 6 vi. y cos 5 vii. y cos 1 iv. ysin 1 48

22 BA01 ENGINEERING MATHEMATICS 01 Eercise 1: i. y sin 4 ii. ycos 1 iii. y tan 1 iv. y sin v. 4 y tan vi. 4 y sin

23 BA01 ENGINEERING MATHEMATICS 01 Eercise : i. y cos ii. y sin iii. sin y 1 sin iv. y sin cos v. tan y vi. y e sin vii. y e viii. y ln i. ln y 50

24 BA01 ENGINEERING MATHEMATICS PARAMETRIC DIFFERENTIATION The implicit of relationship of and y can be epressed in a simpler form by using a third variable, known as the parameter. Eample: Find in terms of the parameter for d 1. t, y t t t d t dt y t t d t 1 d dt * dt d 1 t 1* t t 1 t. t, y 4 4t 4t 51

25 BA01 ENGINEERING MATHEMATICS 01. e t, y sin t Eercise: Find d in terms of the parameter for i. t y t, 1 ii. 5cos t, y 7sin t 1 iii., y 1 t t iv. t v. cos a, y a sin a vi. sin t, y e t t, y t t 5

26 BA01 ENGINEERING MATHEMATICS SECOND DERIVATIVE The second derivative is what you get when you differentiate the derivative. Remember that the derivative of y with respect to is written. The second derivative is written d d y, pronounced "dee two y by d squared". d Eample: Find d and d y d if a) y d

27 BA01 ENGINEERING MATHEMATICS 01 Eercise: Find d and d y if : d f t 45t 11t t i. 1 ii. y iii. y p q 1 v. y 5 iv. vi. f 1 54

28 BA01 ENGINEERING MATHEMATICS 01 POLITEKNIK KOTA BHARU JABATAN MATEMATIK, SAINS DAN KOMPUTER BA 01 ENGINEERING MATHEMATICS PAST YEAR FINAL EXAMINATION QUESTIONS 1) Using the suitable method differentiate the following variables. a) ( ) ( ) b) c) d) ( ) e) ) Differentiate the equation below. a) b) ( ) c) d) e) ( ) f) ( ) ) Derive the equation below: a) ( )( ) b) c) ( ) d) ( ) e) 4) Using the suitable method differentiate the following variables a) b) ( ) c) ( ) d) ( ) e) f) ( ) 5) Differentiate the equation below. a) ( ) b) c) d) e) 6) Derive the equation below: a) b) ( ) c) ( ) d) e) 55

29 BA01 ENGINEERING MATHEMATICS 01 10) Derive the equation below: 7) Find for the following equations a) b) ( )( ) c) d) e) ( ) ( ) f) 8) Find for the following equations a) b) ( ) a) b) c) d) ( )( ) e) ( ) f) ( ) 11) Find the for the parametric d functions given below in terms of t. a) b) y t 5, t t 4 y 5 t, ln t 4 c) c) t, y 4t t d) e) 9) Using the right method, differentiate the functions given. a) ( ) b) ( ) 1) Find the second derivatives for the function a) y 5 1 b) f 5 c) y 4 c) d) ( ) e) 56