Cambridge Additional Mathematics

358 Introduction to differential calculus (Chapter 13)

c If y=

e^2 x

x

then

dy

dx

=

e^2 x(2)x¡e^2 x(1)

x^2

=

e^2 x(2x¡1)

x^2

fquotient ruleg

Example 14 Self Tutor

Find the gradient function foryequal to: a (ex¡1)^3 b

1

p

2 e¡x+1

a y=(ex¡1)^3

=u^3 where u=ex¡ 1

dy

dx

=

dy

du

du

dx

fchain ruleg

=3u^2

du

dx

=3(ex¡1)^2 £ex

=3ex(ex¡1)^2

b y=(2e¡x+1)

¡^12

=u

¡^12

where u=2e¡x+1

dy

dx

=

dy

du

du

dx

fchain ruleg

=¡^12 u

¡^32 du

dx

=¡^12 (2e¡x+1)

¡^32

£ 2 e¡x(¡1)

=e¡x(2e¡x+1)

¡^32

EXERCISE 13I

1 Find the gradient function for f(x) equal to:

a e^4 x b ex+3 c exp(¡ 2 x) d e

x

2

e 2 e

¡x 2

f 1 ¡ 2 e¡x g 4 e

x

(^2) ¡ 3 e¡x h e
x+e¡x
2
i e¡x
2
j e
1
x k 10(1 +e^2 x) l 20(1¡e¡^2 x)
m e^2 x+1 n e
x
(^4) o e^1 ¡^2 x^2 p e¡^0 :^02 x
2 Find the derivative of:
a xex b x^3 e¡x c e
x
x
d x
ex
e x^2 e^3 x f
ex
p
x
g
p
xe¡x h
ex+2
e¡x+1
3 Find the gradient of the tangent to:
a y=(ex+2)^4 at x=0 b y=
1
2 ¡e¡x
at x=0
c y=
p
e^2 x+10at x=ln3.
4 Given f(x)=ekx+x and f^0 (0) =¡ 8 , findk.
5aBy substituting eln 2 for 2 in y=2x, find
dy
dx
.
b Show that if y=bx where b> 0 , b 6 =1, then
dy
dx
=bx£lnb.
6 The tangent to f(x)=x^2 e¡x at point P is horizontal. Find the possible coordinates of P.
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Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_13\358CamAdd_13.cdr Friday, 4 April 2014 5:25:34 PM BRIAN