ENG1113 Tutorial 4 Differentiation Part III

8/4/2019 ENG1113 Tutorial 4 Differentiation Part III

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Taylors University

School of Engineering

Engineering Mathematics I (ENG1113)Dr. Abdulkareem Sh. Mahdi

24/09/2011September 2011

Tutorial4

Tutorial4

Differentiation(PartIII)

Summary:

1) Differentiation of Trigonometric and Inverse Trigonometric Functions:

Basic Trigonometric functions: xxx

cos)(sind

d= , xx

xsin)(cos

d

d= , xx

x

2sec)(tand

d= .

Inverse Trigonometric functions: To find the derivative of inverse trigonometric functions use implicitdifferentiation and the fundamental trigonometric identities. You may need to remember the following two

trigonometric identities 1cossin 22 + xx and xx 22 sectan1 + .

2) Differentiation of Hyperbolic and Inverse Hyperbolic Functions:

Basic Hyperbolic functions: xxx

cosh)(sinhd

d= , xx

xsinh)(cosh

d

d= , xx

x

2hsec)(tanhd

d= .

Inverse Hyperbolic functions: To find the derivative of inverse hyperbolic functions use implicit differentiationand the fundamental hyperbolic identities. You may need to remember the following two hyperbolic

identities 1sinhcosh22

xx and xx22

hsectanh1 .

3) Differentiation of Logarithmic and Exponential Functions: It might sometimes helpful to use the logarithmic laws and

implicit differentiation to differentiate some complicated functions. This also requires remembering some basic

logarithmic laws. For example

Remember!

Ifx

ay = yx alog= also yxxy lnln)ln( += , yxy

xlnlnln =

, xaxa lnln = .

Basic Logarithmic and Exponential Functions: xx eex

=)(d

d,

xx

x

1)(ln

d

d= .

4) Finding Limits and LHpitals Rule: Sometimes we can evaluate the limits of some functions by direct substitution.

But if the direct substitution gives indeterminate forms like 0/0 or /then we can use LHpitals Rule. Hence, if

( )( )

=

or0

0lim

xg

xf

axthen

( )( )

( )( )xgxf

xg

xf

axax

=

limlim . Still have

or

0

0then try

( )( )

( )( )xgxf

xg

xf

axax

=

limlim again.

Instructions to find the limits of functions in form: ( )( )xgxf

axlim

a) Before using LHpitals Rule, you must satisfy yourself and show that direct substitution gives theindeterminate forms 0/0 or /.

b) If it does, you may use the rule, but not otherwise.c) If necessary, you can apply the rule more than once again until a stage is reached where the limits of the

numerator and denominator are not simultaneously zero or infinity.d) When you use LHpitals Rule, you must differentiate the numerator and the denominator separately.

Dont use quotient rule!!

1) Verify each of the following identities:

(a)

cos

sec1

tansin1

+ , (i)

2cos

2sin2sinsin

++ , (ii) ( ) ( ) ++ sinsincossin2 ,

(iii)

tan1

tan21

sincos

sincos

+

+.


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2) Differentiate the following functions with respect tox:

(a) ( )34sin += xy , (b) ( )72cos2 += xy , (c) ( )x

xxf

2cos

sin

= , (d) ( ) xxf 2sin 2=

(i) ( )223sin xxy += , (ii) ( )2cos 34 += xey , (iii) xy sin= , (iv) ( ) ( )27cos72sin 22 +++= xxy ,

(v) x2eccos .

3) Find f (x) for the following functions:

(a) ( ) ( )34sinh += xxf , (b) ( ) ( )72cosh 2 += xxf , (c) ( )x

xxf

cosh

2sinh

= , (d) ( ) xxf 2sinh 2=

(i) ( ) ( )223sinh xxxf += , (ii) ( ) ( )2cosh 34 += xexf , (iii) ( ) xxf sinh= ,

(iv) ( ) ( ) ( )27cosh72sinh 22 +++= xxxf , (v) ( ) xxf 2echcos= .

4) Determine the derivative of each of the following:

(a) xy 1cos

= , (b) xy1tanh

= , (i) xy1tan

= , (ii) xy1cosh

=

5) Differentiate the following:

(a) ( )xy 3sinh 1= , (b) ( )xy tantanh 1= , (i)

=

2

5cosh

1 xy , (ii) 1sinh

21=

xy , (iii) )xey 21cosh = .

6) Find f(x) for the following functions:

(a) ( ) xxf 4ln= , (b) ( ) xxf10

log= , (c) ( ) ( )xexf x 5ln = , (d) ( )23

6x

exf = , (e) ( ) xexf 2sin= ,

(f) ( ) ( )71+= xexf , (g) ( ) ( )4ln 2 += xxf ,

(i) ( ) ( )12log10 = xxf , (ii) ( ) xxf alog= , (iii) ( )xxx exeexf +++=

3532/, (iv) ( ) x

xe

exf 3

3

1 += ,

(v) ( )2

3 xexf = , (vi) ( ) xexf sin2= , (vii) ( ) xexf sin2= , (viii) ( ) 22

6+

=xexf , (ix) ( ) xxxf = .

7) A model of a body falling under gravity with air resistance gives ( )tev 5.0110 = where v(ms-1) is the velocity, and tis time (s). What is the terminal velocity, i.e., )t

te

5.0110lim

?

8) Evaluatex

xx

x 25

34lim

2

2

+

, Show that an attempt to by apply LHpitals Rule here gives a different and incorrect limit of

4. NB: LHpitals Rule can only be used if direct substitution gives the indeterminate forms

or

0

0.

9) Find the limits of the following functions:

(a)

+

x

xx

x 25

34lim

2

2

, (b)

20

sinlim

x

xx

x

, (i)

xx

xx

x sin

tanlim

0, (ii)

+ tttt

1

1

1lim

0.


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10) Find the limits of the following functions:

(a)

+

xx

exx x

x cos4sin

cos32lim

0

, (b)

x

exx

x

coshlim

0

, (c)

30

cossinlim

x

xxx

x

.

(i)

2

32

0 5

cossin21Lim

x

xx

x

, (ii)

30

sintanLim

x

xx

x

, (iii)

xx

xx

x sin

tanLim

0

,

(iv)

x

x

x cos1

cosh1lim

2

0

, (v)

30

sinsinhlim

x

xx

x

, (vi)caxbx

cbxax

x +

+

2

2

lim (where a, b and care constants).

SOME CHALLENGESSOME CHALLENGESSOME CHALLENGESSOME CHALLENGES

C.1)If xy sec= , prove that 42

2

2

d

d

d

dy

x

y

x

yy +

= .

C.2)If tx cosh= and ty sinh= , show that tx

y 32

2

cosechd

d= .

C.3)Ifx

xy 10= , findx

y

d

d. (Ans.: ( )xxxx

x

ln110ln10 + )

C.4)The reability function, R(t), is given by ( ) 1=t

eetR , show that ( )[ ] 11d

d +=

tet

etRt

.

C.5) The efficiency of a thermal cycle is

=

1

2

12

1 ln1T

T

TT

T where T1 and T2T1 are temperatures at specific stages

of the cycle. Assuming that T2 = T1(1 + x), obtain as a function ofx, and deduce the value of lim0x

. (Ans.: 0)

C.6) The responsexof a valve when subjected to a certain input is given by

=

2

tan13

d

d

t

x

t

x(t> 0) where

x 0 as t 0+ (tapproaches 0 from the positive side). What is the limiting value oft

x

d

das t 0+ ? (Ans.:

2

3)

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ENG1113 Tutorial 4 Differentiation Part III