Cambridge Additional Mathematics

Introduction to differential calculus (Chapter 13) 359

Discovery 8 The derivative of lnx

If y=lnx, what is the gradient function?

What to do:

1 Click on the icon to see the graph of y=lnx. Observe the gradient function being

drawn as the point moves from left to right along the graph.

2 Predict a formula for the gradient function of y=lnx.

3 Find the gradient of the tangent to y=lnx for x=0: 25 , 0 : 5 , 1 , 2 , 3 , 4 , and 5.

Do your results confirm your prediction in 2?

From theDiscoveryyou should have observed: If y=lnx then

dy

dx

=

1

x

The proof of this result is beyond the scope of this course.

THE DERIVATIVE OF lnf(x)

Suppose y=lnf(x)

) y=lnu where u=f(x).

Now

dy

dx

=

dy

du

du

dx

fchain ruleg

)

dy

dx

=

1

u

du

dx

=

f^0 (x)

f(x)

Function Derivative

lnx

1

x

lnf(x)

f^0 (x)

f(x)

Example 15 Self Tutor

Find the gradient function of:

a y=ln(kx), ka constant b y=ln(1¡ 3 x) c y=x^3 lnx

a y=ln(kx)

)

dy

dx

=

k

kx

=

1

x

b y=ln(1¡ 3 x)

)

dy

dx

=

¡ 3

1 ¡ 3 x

=

3

3 x¡ 1

c y=x^3 lnx

)

dy

dx

=3x^2 lnx+x^3

³ 1

x

́

fproduct ruleg

=3x^2 lnx+x^2

=x^2 (3 lnx+1)

J Derivatives of logarithmic functions

CALCULUS

DEMO

ln( ) = ln + ln

=ln +

ln( ) ln

kx k x

x

kx x

constant

so and

both have derivative

__

Qv_.

4037 Cambridge

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Y:\HAESE\CAM4037\CamAdd_13\359CamAdd_13.cdr Tuesday, 7 January 2014 9:56:42 AM BRIAN