Cambridge Additional Mathematics

360 Introduction to differential calculus (Chapter 13)

The laws of logarithms can help us to differentiate some logarithmic functions more easily.

For a> 0 , b> 0 , n 2 R: ln(ab)=lna+lnb

ln

³a

b

́

=lna¡lnb

ln(an)=nlna

Example 16 Self Tutor

Differentiate with respect tox:

a y=ln(xe¡x) b y=ln

·

x^2

(x+ 2)(x¡3)

̧

a y=ln(xe¡x)

=lnx+lne¡x fln(ab)=lna+lnbg

=lnx¡x flnea=ag

)

dy

dx

=

1

x

¡ 1

b y=ln

·

x^2

(x+ 2)(x¡3)

̧

=lnx^2 ¡ln[(x+ 2)(x¡3)] fln

³

a

b

́

=lna¡lnbg

=2lnx¡[ln(x+ 2) + ln(x¡3)]

=2lnx¡ln(x+2)¡ln(x¡3)

)

dy

dx

=

2

x

¡

1

x+2

¡

1

x¡ 3

EXERCISE 13J

1 Find the gradient function of:

a y=ln(7x) b y=ln(2x+1) c y=ln(x¡x^2 )

d y=3¡2lnx e y=x^2 lnx f y=

lnx

2 x

g y=exlnx h y=(lnx)^2 i y=

p

lnx

j y=e¡xlnx k y=

p

xln(2x) l y=

2

p

x

lnx

m y=3¡4ln(1¡x) n y=xln(x^2 +1)

2 Find

dy

dx

for:

a y=xln 5 b y=ln(x^3 ) c y=ln(x^4 +x)

d y= ln(10¡ 5 x) e y= [ln(2x+1)]^3 f y=

ln(4x)

x

g y=ln

³

1

x

́

h y=ln(lnx) i y=

1

lnx

A derivative function

will only be valid on

the domain of

the original function.

at most

cyan magenta yellow black

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Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_13\360CamAdd_13.cdr Tuesday, 7 January 2014 9:54:49 AM BRIAN