Cambridge Additional Mathematics

356 Introduction to differential calculus (Chapter 13)

Discovery 7 The derivative of y=bx

The purpose of this Discovery is to observe the nature of the derivatives of f(x)=bx for various

values ofb.

What to do:

x y

dy

dx

dy

dx

¥y

0

0 : 5

1

1 : 5

2

1 Use the software provided to help fill

in the table for y=2x:

2 Repeat 1 for the following functions:

a y=3x b y=5x c y=(0:5)x

3 Use your observations from 1 and 2 to write a statement about the derivative of the general

exponential y=bx for b> 0 , b 6 =1.

From theDiscoveryyou should have found that:

If f(x)=bx then f^0 (x)=f^0 (0)£bx.

Proof:

If f(x)=bx,

then f^0 (x) = lim

h! 0

bx+h¡bx

h

ffirst principles definition of the derivativeg

= lim

h! 0

bx(bh¡1)

h

=bx£

μ

lim

h! 0

bh¡ 1

h

¶

fas bx is independent ofhg

But f^0 (0) = lim

h! 0

f(0 +h)¡f(0)

h

= lim

h! 0

bh¡ 1

h

) f^0 (x)=bx£f^0 (0)

Given this result, if we can find a value ofbsuch that f^0 (0) = 1, then we will have founda function which

is its own derivative!

CALCULUS

DEMO

y

x

gradient isf (0)'

O

y=bx

cyan magenta yellow black

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Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_13\356CamAdd_13.cdr Friday, 4 April 2014 5:25:26 PM BRIAN