basic derivatives M a t h e m a t i c a l D i f f e r e n t i a t i o n f o r P o l y t e c h n i c S t u d e n t s N O O R H I D A Y A H J A M A L U D I N
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BASIC DERIVATIVES Editor Noor Hidayah binti Jamaludin Writer Noor Hidayah binti Jamaludin Designer Noor Hidayah binti Jamaludin 1st Edition 2022 All right reserved. No part of this work may be reproduced, stored in any form or by any means without the written permission of the publisher. Published by Politeknik Muadzam Shah Lebuhraya Tun Abdul Razak 26700 Muadzam Shah Pahang Darul Makmur Tel : 09-450 2002/ 2006/ 2007 Fax: 09-450 2009
PREFACE Alhamdulillah, all praise and gratitude to Allah SWT for this book have been completed. Basic Derivatives book are designed to be an additional reference for polytechnic engineering students who are taking Engineering Mathematics 2 courses. This topic have been carefully arranged to conform to the syllabus used for Engineering Mathematics 2 course. In addition, sample questions and answers are also provided to help students to understand better. We hope that this book will be useful for students in enhancing their mathematical skills and develop their interest towards mathematics. Thank you.
TABLE OF CONTENTS UNDERSTAND DERIVATIVES 1 2 2 3 5 8 10 13 DERIVATIVE OF A BASIC FUNCTION Derivative of a constant function Derivative of an identity function Derivative of power function Derivative of sum and difference function TECHNIQUES OF DIFFERENTIATION Product rule Quotient rule Chain rule DERIVATIVE OF EXPONENTIAL FUNCTION DERIVATIVE OF LOGARITHMIC FUNCTION DERIVATIVE OF TRIGONOMETRIC FUNCTION SECOND ORDER OF DIFFERENTIATION DERIVATIVE PARAMETRIC EQUATION IMPLICIT DIFFERENTIATION 16 18 21 26 28 30 PREFACE
1 ➢ Differentiation is defined in mathematics as the derivative of a function with respect to an independent variable. ➢ In other words, differentiation is the process of finding derivatives. ➢ The derivative is shown by the notations dy dx or f x ( ) when a function is given as y = f(x). ➢ The most important Differentiation formulas are included in the table below. Derivative of function Rule/formula Constant ( ) = 0 d k dx Identity ( ) 0 d ax dx = Power ( ) d n n 1 ax nax dx − = Derivative of Sum and Difference ( ) ( ) ( ) ( ) d d d f x g x f x g x dx dx dx = Exponential ( ) d x x e e dx = ( ) d du u u e e dx dx = • Logarithmic ( ) 1 ln d x dx x = ( ) 1 ln d du u dx u dx = • Trigonometric (sin cos ) d x x dx = (cos sin ) d x x dx = − ( ) 2 tan sec d x x dx = (sin cos ) d du u u dx dx = • (cos sin ) d du u u dx dx = − • ( ) 2 tan sec d du u u dx dx = • Product rule ( ) d dv du u v u v dx dx dx • = + Quotient rule 2 du dv v u d u dx dx dx v v − = Chain rule dy dy du dx du dx = Parametric equation dy dy dt dx dt dx =
2 1.1 Derivative of a constant function 1.2 Derivative of an identity function Let k be a constant, if f x k then f x ( ) = = , 0 ( ) We also can write as ( ) = 0 d k dx Find the derivative for the following: a) y = 3 0 dy dx = b) y = 5 0 dy dx = c) y = 6 0 dy dx = d) 1 2 y = 0 dy dx = e) y = −4.56 0 dy dx = f) 9 2 y = 0 dy dx = Example 1 If k is a constant, then ( ) d k x k dx = Find the derivative for the following: a) y x = 3 3 dy dx = b) y x = 7 7 dy dx = Example 2
3 1.3 Derivative of Power function If a is a constant and n ia a positive integer, then ( ) d n n 1 ax nax dx − = Find the derivative for the following functions. a) 2 y x = ( ) 2 1 2 2 dy x dx x − = = b) 3 y x = 7 ( ) 3 1 2 3 7 21 dy x dx x − = = c) 3 4 2 y x = ( ) 3 3 3 4 2 6 dy x dx x = = d) 2 2 5 y x = ( ) 2 1 3 2 2 5 4 5 dy x dx x − − − = − − = e) 4 2 3 y x − = ( ) 3 3 4 2 3 8 3 dy x dx x − − = − − = f) 3 4 5 y x − = ( ) 3 2 2 4 5 4 3 5 12 5 y x dy x dx x = = = Example 3 c) y x = 10 3 2 dy dx = − d) 3 2 y x = − 3 2 dy dx = − e) y x = 4.76 4.76 dy dx = f) 3 2 y x = 3 2 dy dx =
4 g) ( ) 3 5 f x x = ( ) ( ) 53 23 23 53 53 f x x f x x x = = = h) 2 2 5 y x = ( ) 2 1 32 2 5 45 dy x dx x − − − = −− = i) 4 2 3 y x − = ( ) 3 34 2 3 83 dy x dx x − − = −− = j) 3 4 5 y x − = ( )3 2 2 45 4 3 5 125 y x dy x dx x = == k) ( ) 1 f x x = ( ) ( ) 12 32 32 1212 f x x f x x x − − − = − = − = l) 233 y x = 23 53 53 3 23 23 y x dy x dx x − − − = = − − =
5 1.4 Derivative of Sum and Difference function I If f and g are both differentiable, then ( ) ( ) ( ) ( ) d d d f x g x f x g x dx dx dx = Find the derivative for the following functions. a) 2 y x = −5 2 dy x dx = − b) 4 2 2 3 8 5 y x x x = − + ( ) ( ) 2 4 3 3 3 3 2 3 8 5 2 2 4 3 8 5 1 12 8 5 y x x x dy x x dx x x − − − = − + = − − + = − + c) ( ) 2 2 3 3 f x x4 x − = + ( ) ( ) ( ) ( ) 2 2 3 2 3 2 1 3 1 3 3 4 3 4 2 3 2 4 3 2 8 f x x x f x x x f x x x x x − − − = + = + = + = + d) 3 y x = + 7 2 ( ) 2 2 7 0 7 dy x dx x = + = e) 4 3 2 5 1 3 y x x = + − ( ) ( ) 2 2 4 3 2 5 0 3 4 10 dy x x dx x x = + − = + f) ( ) 5 3 4 4 5 f x x x x − − = + + ( ) ( ) ( ) ( ) 1 3 5 2 1 2 6 2 1 2 6 2 4 4 5 4 1 3 5 4 5 2 12 1 20 5 2 f x x x x f x x x x x x x − − − − − = + + = + − + = − + Example 4
6 g) 3 2 5 2 2 y x x = + − 3 1 2 5 0 2 3 5 dy x dx x = + − = + i) 2 y x x = + − 5 9 2 ( ) 1 2 2 12 12 5 9 21 2 5 9 0 2 9 10 2 y x x dy x x dx x x − − = + − = + − = + k) 3 6 x x 6 y x+ = ( ) ( ) 3 6 6 3 5 3 5 4 6 4 6 6 1 66 3 5 6 3 30 x x y x x x x x x dy x x dx x x − − − − − − = + = + = + = − + − = − − h) ( ) 3 2 5 5 2 4 2 f x x x x − = + + + ( ) ( ) ( ) ( ) 2 3 3 4 3 4 5 5 2 4 2 5 2 3 5 2 0 2 5 15 2 f x x x x f x x x x x − − − − − − = + + + = − + − + + = − − + j) 12 y x x 6 = + 32 12 126 3 6 23 6 2 y x x dy x dx x = + = + = +
7 1. Find the derivative for the following functions: a) y = 3 e) 3 5 y = b) y = −2.78 f) y =14 c) y = 8 g) 6 5 y = − 2. Find the derivative for the following functions: a) 4 y x = 5 b) 5 y x = 3 c) 4 9 7 y x = d) 3 8 5 y x − = e) 7 2 7 y x = f) 2 6 5 y x − = − g) 3 5 4 15 x − − h) 2 y x = − 3. Find the derivative for the following functions. a) 2 y x x = + + 2 5 b) 4 y x x = − + 3 6 9 c) y x = 4 d) 3 2 y x x x 5 8 − = − + e) 3 y x x x = + − 25 15 f) 1 2 2 1 6 8 2 y x x − = + + g) 2 2 8 y x3 x = − h) 3 5 2 5 y x x = + Practice Problems 1
8 In this subtopic, we discuss 3 techniques in solving differentiation problems which are product rule, quotient rule and chain rule. Find the derivative for the following functions: a) ( )( ) 3 y x x = + − 4 3 3 2 ( ) 3 2 Let 4 3 12 u x du x dx = + = (3 2) 3 v x dv dx = − = Substitute into the formula, ( )( ) ( )( ) 3 2 3 3 2 3 2 4 3 3 3 2 12 12 9 36 24 48 24 9 dy dv du u v dx dx dx x x x x x x x x = + = + + − = + + − = − + b) ( )( ) 3 2 y x x = − + 2 4 6 ( ) 3 2 Let 2 4 6 u x du x dx = − = ( ) 2 6 2 v x dv x dx = + = Substitute into the formula, ( )( ) ( )( ) 3 2 2 4 4 2 4 2 2 4 2 6 6 4 8 6 36 10 36 8 dy dv du u v dx dx dx x x x x x x x x x x x = + = − + + = − + + = + − Example 5 If u and v are both differentiable, then ( ) d dv du u v u v dx dx dx • = + Product Rule
9 c) ( )( ) 10 2 y x x = + − 3 3 ( ) 2 Let 3 2 u x du x dx = + = ( ) ( ) 10 9 3 10 3 v x dv x dx = − = − Substitute into the formula, ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) 9 10 2 9 2 9 2 2 9 2 3 10 3 3 2 3 3 10 3 2 3 10 30 2 6 3 12 30 6 dy dv du u v dx dx dx x x x x x x x x x x x x x x x = + = + − + − = − + + − = − + + − = − + − c) ( )( ) 3 2 y x x = − + 2 4 6 ( ) 3 2 Let 2 4 6 u x du x dx = − = ( ) 2 6 2 v x dv x dx = + = Substitute into the formula, ( )( ) ( )( ) 3 2 2 4 4 2 4 2 2 4 2 6 6 4 8 6 36 10 36 8 dy dv du u v dx dx dx x x x x x x x x x x x = + = − + + = − + + = + −
10 Find the derivative for the following functions. a) 2 6 2 x y x = − 2 Let 6 12 u x du x dx = = 2 1 v x dv dx = − = − substitute into the formula, ( )( ) ( )( ) ( ) ( ) 2 2 2 2 2 2 12 6 1 2 24 6 2 du dv v u dy dx dx dx v x x x x x x x − = − − − = − − = − b) ( ) 2 3 1 5 2 x y x + = − 2 Let 3 1 6 u x du x dx = + = 5 2 2 v x dv dx = − = − substitute into the formula, ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 2 2 5 2 6 3 1 2 5 2 30 12 6 2 5 2 30 12 6 2 5 2 18 6 2 5 2 du dv v u dy dx dx dx v x x x x x x x x x x x x x x − = − − + − = − − − − − = − − + + = − + + = − Example 6 If u and v are both differentiable, then 2 du dv v u d u dx dx dx v v − = Quotient Rule
11 c) ( ) ( ) 2 3 6 2 5 x y x + = − ( ) ( ) ( ) ( ) 2 2 1 Let 6 2 2 6 2 2 4 6 2 u x du x dx x − = + = + = + ( ) ( ) ( ) ( ) 3 3 1 2 5 3 5 1 3 5 v x dv x dx x − = − = − = − Substitute into the formula, ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( ) 2 3 2 2 2 3 2 2 2 6 2 6 4 5 4 6 2 6 2 3 5 5 6 2 5 4 5 3 6 2 2 6 2 5 4 20 18 6 5 6 2 5 ( 38 2 ) 5 6 2 ( 38 2 ) 5 du dv v u dy dx dx dx v x x x x x x x x x x x x x x x x x x x x x x − = − + − + − = − + − − − + = − + − − − − = − + − − − = − + − − = − d) ( ) ( ) 2 4 3 4 x y x − = + ( ) ( ) ( ) ( ) 2 2 1 Let 3 2 3 1 2 3 u x du x dx x − = − = − = − ( ) ( ) ( ) ( ) 2 2 1 4 2 4 1 2 4 v x dv x dx x − = + = + = + Substitute into the formula, ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) ( ) ( )( )( ) ( ) ( ) ( ) 2 2 2 2 4 8 8 8 7 4 2 3 3 2 4 4 3 4 2 4 2 3 4 3 4 2 8 2 6 4 3 4 14 4 14 3 4 du dv v u dy dx dx dx v x x x x x x x x x x x x x x x x x x x x − = + − − − + = + − + + − − = + − + + − + = + − + = + − = +
12 1. Differentiate the following functions. a) ( ) ( ) 2 y x x = − + 2 3 1 b) ( )( ) 4 2 y x x x = + − 7 5 c) ( )( ) 2 y x x = + + 3 5 7 d) ( )( ) 2 y x x = + − 3 5 2 3 2. Differentiate each of the following functions. a) 3 6 2 7 x y x + = − b) ( ) 3 2 2 2 1 1 x y x + = − c) ( ) ( ) 2 3 6 3 4 x y x + = − d) ( ) 3 7 6 4 3 x y x + = − e) ( ) 2 2 3 3 x y x − = + Practice Problems 2
13 Find the derivative for the following functions by using chain rule. a) ( ) 2 y x = + 2 2 2 y u dy u du = = 2 1 u x du dx = + = Substitute into the formula, ( ) 2 1 2 2 2 dy dy du dx du dx u u x = = = = + b) ( ) 3 y x 6 2 − = − 3 4 6 18 y u dy u du − − = = − 2 1 u x du dx = − = Substitute into the formula, ( ) 4 4 4 18 1 18 18 2 dy dy du dx du dx u u x − − − = = − = − = − − Example 6 The chain rule allows us to calculate the derivative of composite functions. The formula is dy dy du dx du dx = Chain Rule
14 c) ( ) 3 3 2 4 1 4 y x = + ( ) 3 2 2 3 4 3 3 4 9 4 y u dy u du u = = = 2 4 1 8 u x du x dx = + = Substitute into the formula, ( ) 2 2 2 2 9 8 4 18 18 4 1 dy dy du dx du dx u x u x = = = = + d) ( ) 5 3 3 4 y x = − ( ) 5 y x 3 3 4 − = − 5 6 3 15 y u dy u du − − = = − 3 4 3 u x du dx = − = Substitute into the formula, ( ) 5 5 5 15 3 75 75 3 4 dy dy du dx du dx u u x − − − = = − = − = − − Example 6
15 Differentiate the following functions by using chain rule. a) b) ( ) 3 y x = +1 b) ( ) 2 2 1 3 1 x y = + c) ( ) 2 3 y x = + 2 d) ( ) 3 3 4 y x = + 4 5 e) ( ) 3 5 2 3 y x x = + f) 2 y x x = − 7 Practice Problems 3
16 To differentiate exponential function, we need to consider two rules which are basic rule and generalized rule. Find the derivative for the following functions. a) 4 x y e = 4 dy x e dx = b) ( ) 7 3 4 x f x e = ( ) ( ) ( ) 3 3 7 2 2 7 4 21 84 x x f x e x f x x e = = c) 2x y e = ( ) 2 2 2 2 x x dy e dx dy e dx = = d) 4 3 x y e = ( ) 4 4 3 4 12 x x dy e dx dy e dx = = e) ( ) 5 2 6 x f x e− = ( ) ( ) 5 5 2 4 4 2 6 10 60 x x f x e x x e − − = − = − f) ( ) 3 4 2 5 x f x e− + = − ( ) ( ) ( ) ( ) 4 2 4 2 4 2 3 4 5 3 4 5 12 5 x x x f x e f x e f x e − + − + − + = − − = = Example 7 Basic rule: ( ) d x x e e dx = Generalized rule: ( ) d du u u e e dx dx = •
17 1. Differentiate the following exponential functions. a) 5x y e − = b) 5 3 2 x y e = c) ( ) 3 10 x f x e− = − d) ( ) 6 7 3 x f x e + = − e) 3 7 x y e − = f) 4 3 2 x y e − = Practice Problems 4
18 To differentiate exponential function, we consider two rules which are basic rule and generalized rule. Find the derivative for the following functions. a) y x = 4ln 1 4 4 dy dx x x = = b) 3 ln 5 y x = 3 1 5 3 5 dy dx x x = = c) y x = −7ln 1 7 7 dy dx x x = − = d) y x = ln9 ln 9 1 9 9 1 y x dy dx x x = = • = e) y x = 5ln 6 5ln 6 1 5 6 6 5 y x dy dx x x = = • = f) 3 ln 7 4 y x = 3 ln 7 4 3 1 7 4 7 3 4 y x dy dx x x = = • = Example 8 Basic rule: 1 ln d x dx x = Generalized rule: d du 1 1 dx u u dx = •
19 g) y x = −7ln10 7ln101 7 10 10 7 y x dy dx x x = − = − • − = h) ( ) 3 f x x = ln8 ( ) 2 3 1 24 83 f x x x x = • = i) f x x ( ) = − ln 7 5 ( ) ( ) 1 7 7 5 7 7 5 f x xx = • − = − j) ( ) 3 3 ln 7 4 4 y x x = + ( ) ( ) ( ) ( )3 2 3 23 3 ln 7 4 4 3 1 7 12 4 7 4 3 7 12 4 7 4 y x x dy x dx x xx x x = + = • + + + = + k) ( ) ( ) 7 f x x ln 2 − = ( ) ( ) ( ) ( ) 7 8 7 7 8 ln 21 14 2 1 14 27 f x x f x x xx x x − − − = = • −− = • − = l) ( ) ( ) 2 f x x = − − 7ln 4 8 ( ) ( ) ( ) ( ) 2 2 2 7ln 4 8 1 7 8 4 8 56 4 8 f x x f x x xx x = − − = − • − − = − m) ( ) 7 ln 10 6 5 y x = − ( ) 7 ln 10 6 5 7 1 10 5 10 6 14 10 6 y m dy dm m m = − = • − = − n) ( ) 4 y k = − 7ln 3 6 ( ) 4 3 43 4 3 4 7ln 3 6 1 7 12 3 6 84 3 6 28 2 w k dw k dk kk k k k = − = • − = − = −
20 Differentiate the following functions. ( ) ( ) ( ) ( ) 5 a) 4ln 3 b) ln 5 c) 7ln d) ln 3 e) ln 5 2 f) 4ln 10 12 g) ln 5 1 h) ln 6 4 2 y x y x y x y x y x y x y x y x − = = = − = = − = + = = − Practice Problems 5
21 In this subtopic, we just consider three basics trigonometric functions which are sine, cosine and tangent. To differentiate trigonometric function, we consider two rules which are basic rule and generalized rule as stated in the table below. Function Basic rule Generalized rule Sine (sin cos ) d x x dx = (sin cos ) d du u u dx dx = • Cosine (cos sin ) d x x dx = − (cos sin ) d du u u dx dx = − • Tangent ( ) 2 tan sec d x x dx = ( ) 2 tan sec d du u x dx dx = • Find dy/dx for the following functions. a) y x = sin 5 cos5 5( ) 5cos5 dy x dx x = = b) y x = sin 8 cos8 8 cossin 8 dy x dx x = • = c) y x = 5sin 10 ( ) ( ) ( ) 5cos 10 10 50cos 10 dy x dx x = • = d) ( ) 3 sin 4 2 y x = ( ) ( ) 3 cos 4 4 2 6cos 4 dy x dx x = • = e) y x = + sin 3 5 ( ) ( ) ( ) cos 3 5 3 3cos 3 5 dy x dx x = + • = + f) ( ) 2 y x = + sin 2 7 ( ) ( ) 2 2 cos 2 7 4 4 cos 2 7 dy x x dx x x = + • = + Example 9
22 Find dy/dx for the following functions. a) y x = 4cos 4 sin ( ) 4sin dy x dx x = − = − b) y x = 8cos 8 sin ( ) 8sin dy x dx x = − = − c) y x = cos 12 ( ) ( ) ( ) sin 12 12 12sin 12 dy x dx x = − • = − d) y x = 8cos 8( ) ( ) ( ) 8 sin 8 8 512sin 8 dy x dx x = − • = − e) 3 y x = cos3 ( ) ( ) 3 2 2 3 sin 3 9 9 sin 3 dy x x dx x x = − • = − f) y x = + cos 5 2 ( ) ( ) ( ) sin 5 2 5 5sin 5 2 dy x dx x = − + • = − + g) ( ) 1 2 cos 4 5 6 y x = + ( ) ( ) 2 2 1 sin 4 5 8 6 4 sin 4 5 3 dy x x dx x = − + • = − + h) 1 5cos 10 4 y x = − 1 1 5 sin 10 4 4 5 1 sin 10 4 4 dy x dx x = − − • − = − Example 10
23 Find dy/dx for the following functions. a) y x = 3tan 2 3sec dy x dx = b) y x = 6tan 2 6sec dy x dx = c) y x = tan 14 ( ) ( ) 2 sec 14 dy x dx = d) y x = 3tan 8( ) ( ) ( ) ( ) 2 2 3tan 8 3sec 8 8 24sec 8 y x dy x dx x = = • = e) ( ) 3 y x = tan 2 ( ) ( ) 2 3 2 2 2 3 sec 2 6 6 sec 2 dy x x dx x x = • = f) y x = + tan 4 2 ( ) ( ) ( ) 2 2 sec 4 2 4 4sec 4 2 dy x dx x = + • = + g) ( ) 1 2 tan 2 4 3 y x = + ( ) ( ) 2 2 2 2 1 sec 2 4 4 3 4 sec 2 4 3 dy x x dx x x = + • = + h) 2 3tan 4 7 y x = − 2 2 2 2 3 sec 4 7 7 6 2 sec 4 7 7 dy x dx x = − • = − Example 11
24 Find dy/dx for the following functions. a) y x x = + sin cos cos sin dy x x dx = − b) y x x = − sin 4 cos5 cos 4 4 sin 5 5 ( ) ( )( ) 4cos 4 5sin 5 dy x x dx x x = − − = + c) y x x = − 5sin 4 3cos 2 5cos 4 4 3 sin 2 2 ( ) ( )( ) 20cos 4 6sin 2 dy x x dx x x = − − = + d) 1 sin 3 5tan 3 4 y x x = + ( ) ( )( ) 2 2 1 cos3 3 5 sec 5 3 4 3 cos3 15sec 5 4 dy x x dx x x = + = − e) ( ) 2 y x x = + − 5 3cos 3 12 ( ) ( ) ( ) 2 5 3 sin 3 12 3 10 9sin 3 12 dy x x dx x x = + − − • = − − f) y x s x = + − 3 4 in 3 2 ( ) ( ) ( ) ( ) 1 2 1 2 1 2 3 4sin 3 2 1 4 cos 3 2 3 2 1 12cos 3 2 2 y x x dy x x dx x x − − = + − = + − • = + − g) ( ) 2 2 13 y x 3tan 5 4 x = − + ( ) ( ) ( ) ( ) 2 2 2 2 3 2 2 3 3tan 5 4 13 3 sec 5 4 10 2 13 30 sec 5 4 26 y x x dy x x x dx x x x − − − = − + = − • + − = − − h) 3 1 3 in 5 4 x y e s x = + − 3 3 1 3 3 cos 5 5 4 1 3 15cos 5 4 x x dy e x dx e x = • + − • = + − Example 12
25 Find the derivative of the following trigonometric functions. a) y x = sin 5 b) y x = 7sin c) y x = 4sin 2( ) d) y x = 4cos e) 1 cos 6 y x = f) y x = cos 7 g) y x = − cos 11 3 ( ) h) y x = 3tan i) tan 4 x y = j) 3 y x = sin 7 k) 4 tan 2 5 y x = l) 4 y x = −cos6 m) y x = − − cos 6 4 ( ) n) ( ) 3 y x x = + 4sin 3 10 o) ( ) 2 y x x = − 2cos 4 14 p) ( ) 2 y x = − − 3 sin 1 q) y x x = + cos5 2sin 3 r) ( ) ( ) 2 tan 2 2 5sin 2 4 3 y x x = − + + s) ( ) ( ) 2 y x x = + − − tan 5 4 cos 4 2 t) ( ) 3 y x = − + 3 4cos 3 2 u) y x x x = + − + 2 2sin 7 cos 3 1 ( ) v) cos4 x y e x = + w) 4 cos3 x y e x = + x) 3 2cos 2 x y e x = + y) 2 sin3 x y x e − = − z) ( ) 6 sin 4 1 x y x e = − + Practice Problems 6
26 Second order of differentiation is the derivative of the first derivative of a function. The second derivative of a function y usually denoted as 2 2 d y dx . Differentiate the following functions. a) 5 y x x = + 4 7 5 4 2 3 2 3 4 7 20 7 80 0 80 dy x x dx x d y x dx x = + = + = + = b) 3 4 y x x x 2 5 3 − = + + 2 5 2 6 2 6 6 20 3 12 100 0 12 100 dy x x dx d y x x dx x x − − − = − + = + + = + c) 5 y x x x = − + 4 5 sin3 ( ) 4 4 20 5 cos 3 3 20 5 3cos 3 dy x x dx x x = − + = − + ( )( ) ( ) 2 3 2 3 80 0 3 sin 3 3 80 9 sin 3 d y x x dx x x = − + − = − c) y x = 5cos ( ) ( ) ( ) 2 2 5 sin 5 sin 5 cos dy x dx x d y x dx = − = − = − d) y x = 5cos7 ( )( ) ( ) ( )( ) ( ) 2 2 5 sin 7 7 35 sin 7 35 cos 7 7 245 cos 7 dy x dx x d y x dx x = − == − = − = − e) ( ) 2 y x = + 2 5 ( ) ( ) ( ) 2 1 2 2 2 2 5 2 4 2 5 8 20 8 0 8 dy x dx x x d y dx − = + = + = + = + = Example 10
27 Find the second derivative of the following functions. a) 4 5 y x x x 2 7 2 − = − + b) 4 3 15 3 x y x e − = + c) y x x = + 5cos sin 2 d) ( ) 3 f x x x 4cos 16 − = − e) y x = 7 sin 7 f) ( ) 6 y x 7 5 − = − g) ( ) 3 y x = − 5 2 h) ( ) 9 2 3 5 x f x e x = − Practice Problems 7
28 ➢ When x and y are expressed in terms of a third variable it is called a parameter. ➢ To differentiate parametric equations to find derivative of y with respect to x, we use the chain rule method. a) Find dy dx when x t = 6 and 2 y t = 3 . Solution 6 6 x t dx dt = = 2 3 6 x t dy t dt = = Subtitute into the formula, 1 6 6 dy dy dt dx dt dx t t = = = b) Find dy dx when 3 5 t x e = and 2 y t t = − 3 2 . Solution 3 3 3 5 5 3 15 t t t x e dx e dt e = = • = 2 3 2 6 2 y t t dy t dt = − = − Subtitute into the formula, ( ) 3 3 1 6 2 15 6 2 15 t t dy dy dt dx dt dx t e t e = = − − = c) Example 19 When the parameter in the equations is “t”, the chain rule is defined as: dy dy dt dx dt dx =
29 Find dy dx when x t = cos and y t = sin . Solution cos sin 1 sin x t dx t dt dt dx t = = − = − sin cos x t dy t dt = = By using the chain rule: 1 cos sin cos sin dy dy dt dx dt dx t t t t = = − = − Example 19 For each of the following functions, determine dy dx . a) 2 3 x t y t = + = − 1, 1 b) x t y t = = 3cos , 3sin c) x t t y t t = + = − , d) 3 2 x t y t t = + = + 2 1, cos e) 2 , 2 1 t x te y t − = = + Practice Problems 8
30 ➢ Implicit differentiation is a powerful technique to find an instantaneous rate of change dy dx when there is an equation relating x and y . ➢ In implicit differentiation, we differentiate each side of an equation with two variables (usually x and y ) by treating one of the variables as a function of the other. Find dy dx for all of the following functions. a) 2 3 y x x = − 7 ( ) ( ) ( ) 2 3 2 2 7 2 3 7 3 7 2 d d d y x x dx dx dx dy y x dx dy x dx y = − = − − = b) xy = 2 ( ) ( ) ( ) ( ) 2 1 1 0 0 d d xy dx dx dy y x dx dy y x dx dy x y dx dy y dx x = + = + = = − − = Example 22
31 c) 2 3 x y xy − =10 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 3 3 2 2 3 2 2 2 2 3 2 2 3 3 2 2 10 2 3 1 1 0 2 3 0 3 2 3 223 d d d x y xy dx dx dx dy dy x y x y y x dx dx dy dy xy x y y x dx dx dy dy x y x y xy dx dx dy x y x y xy dx dy y xy dx x x y − = + − + = + − − = − = − − = −− = − d) 3 2 2 x y x x y + − = 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 2 2 1 3 2 2 2 2 3 2 2 3 2 3 2 d d d d x y x x y dx dx dx dx dy dy x y x y x dx dx dy dy x y x y x dx dx dy dy y x x y x dx dx dy y x x y x dx dy x y x dx y x + − = + − = + + − = + − = + − − = + − + − = − e) y x y = + sin cos ( ) ( ) ( ) ( ) ( ) ( ) sin cos sin cos 1 cos sin cos sin sin cos 1 sin cos cos 1 sin y x y d d d y x y dx dx dx dy dy x y dx dx dy dy x y dx dx dy dy y x dx dx dy y x dx dy x dx y = + = + = + − = − + = + == +
32 For each of the following functions, determine dy dx . a) 2 2 x y − = 5 b) 2 2 x xy y + + = − 3 1 c) 2 3 5 10 x x y − = d) 2 sin 4 cos y x y x + + = e) 2 y x xy sin 3 9 + = f) 2 tan5 sin 3 9 y y x xy − + = Practice Problems 9
33 Practice problems 1 Question 1 a) 0 dy dx = e) 0 dy dx = b) 0 dy dx = f) 0 dy dx = c) 0 dy dx = g) 0 dy dx = Question 2 a) 3 20 dy x dx = b) 4 15 dy x dx = c) 8 367 d dxy x = d) 24 9 5 dy dx x − = − e) 8 2 x dy dx − = − f) 125 dy x dx = − g) 85 4 25 dyx x d = h) 32 dy 1 dx x = Question 3 a) 2 2 dy x dx = + b) 3 12 6 d dx x y = − c) dy 2 dx x = d) 2 32 15 8 x dd x yx = − − e) 2 1 75 15 2 dyx x d x = + + f) 52 32 3 x dxy x d − + = g) 3 16 6 x x dy dx = + h) 4 6 6 25 x dd x yx = − +
34 Practice Problems 2 Question 1 a) (2 3 4 9 )( ) dy x x dx = − − − b) ( )( ) ( )( ) 4 2 3 7 2 5 4 7 dy x x x x x dx = + + − + c) 2 9 42 5 dy x x dx = + + d) ( )( ) ( )( ) 2 3 5 4 2 3 3 dy x x x dx = + + − Question 2 a) ( ) 3 2 2 4 21 12 2 7 dy x x dx x − − = − b) ( ) 4 3 2 3 2 8 6 6 4 1 dy x x x x dx x − + − − = − c) ( ) ( ) 2 3 6 3 4 x y x + = − d) ( ) 4 42 26 4 3 dy x dx x − = − e) ( ) 2 2 2 2 6 6 3 dy x x dx x − + + = +
35 Practice Problems 3 Practice Problems 4 a) ( ) 2 3 1 dy x dx = + b) ( ) 3 2 2 3 1 dy x dx − = − + c) ( ) 2 3 6 2 dy x x dx = + d) ( ) 1 2 3 4 9 5 dy x x dx − = + e) ( )( ) 2 2 3 15 2 1 2 3 dy x x x dx − = − + + f) ( )( ) 1 2 2 1 2 7 7 2 dy x x x dx − = − − a) 5 5 dy x e dx − = − b) 15 3 2 d x dx e y = c) ( ) 3 30 x f x e − = d) ( ) 6 7 18 x f x e + = − e) 3 2 21 dy x dx e x − = − f) 4 3 5 24 x dy dx e x = −
36 Practice Problems 5 4 a) b) c) d) e) f) g) h) 3 5 7 1 5 5 2 20 5 6 5 3 6 4 dy dx x dy dx dy dx dy dx dy dx dy dx dy dx x x x x x x dy dx x − = = = = = = = = − + − −
37 Practice Problems 6 a) 5cos5 dy x dx = b) 7 cos dy x dx = c) 4sin 2( ) dy x dx = d) 4sin dy x dx = − e) 1 sin 6 dy x dx = f) 7sin 7 dy x dx = − g) 11sin 11 3 ( ) dy x dx = − − h) 2 3sec dy x dx = i) 1 2 sec 4 4 dy x dx = j) 2 3 y x x = 21 cos7 k) 8 2 sec 2 5 y x = l) ( ) 3 4 144 sin 6 x dy dx = x m) 6sin 6 4 ( ) dy x dx = − n) ( ) ( ) 2 3 4 3 30 cos 3 10 d x x x x y d = + + o) ( ) ( ) 2 2 8 14 sin 4 14 dy x x x dx = − − − p) ( ) 2 2 cos 1 x dy dx = − x − q) 5sin 5 3 ( x x ) 6cos( ) dy dx = + r) ( ) ( ) 4 2 sec 2 2 10cos 2 4 3 x dy dx = − + + x s) ( ) ( ) 2 2 10 sec 5 4 4sin 4 2 dy x x x dx = + + − t) ( ) 2 3 36 sin 3 2 x dy x x d = + u) 2 14cos 7 3sin 3 1 ( x) ( ) x x dy d = + + + v) 4sin 4( ) x e d x y dx = − w) 4 3sin 3( ) dy x e x dx = − x) 3 4sin 2( ) dy x e x dx = − y) ( ) 2 2 cos 3 2 sin 3 3 x x y x e dy d e x x − − = − = − z) ( ) ( ) 6 6 1 cos sin 4 1 6 4 1 4 x x y x e dy dx x e − + = − + =
38 Practice Problems 7 Practice Problems 8 a) 2 2 2 7 210 24x d x y dx = − b) 2 2 3 6 300 27 d y x dx x = + e c) 2 2 5cos 4sin 2 x x x d y d = − − d) 2 2 5 192 4cos x d x f dx = − − e) 2 2 343sin 7 d y x dx = f) ( ) 2 2 8 2058 7 5 d dx x y = − g) ( ) 2 2 150 5 2 d y dx = x − h) 2 2 9 243 10 x e d f dx = − a) 3 2 dy t dx = b) cos @ cot sin dy t t dx t = − − c) 2 1 2 1 t t dy dx = + − d) 2 6 2 sin dy t dx t t = − e) (1 ) 4 t dy e t dx t − − =
39 Practice Problems 9 a) dy x dx y = b) 2 3 3 2 x x y dy x y d − − + = c) 2 dy 3 10 x d y x x − = − d) sin 2 cos 4 dy x dx x y = − − + e) 2 cos 3 sin 6 dy y x dx y x xy − − + = f) 2 2 3 cos 5sec si 5 n 6 dy y y x y x x d xy − + + = −
40 Anton, H., Bivens, I. C., & Davis, S. (2021). Calculus. John Wiley & Sons. Polanco, C. (2019). Advanced Calculus: Fundamentals of Mathematics. Bentham Science Publishers. Vaquero, J. M., Carr, M., Dios, A. Q., & Richtarikova, D. (Eds.). (2020). Calculus for Engineering Students: Fundamentals, Real Problems, and Computers. Academic Press.