Introduction to differential calculus (Chapter 13) 355

EXERCISE 13H

1 Use the quotient rule to find

dy

dx

if:

a y=

1+3x

2 ¡x

b y=

x^2

2 x+1

c y=

x

x^2 ¡ 3

d y=

p

x

1 ¡ 2 x

e y= x

(^2) ¡ 3
3 x¡x^2
f y=p x
1 ¡ 3 x
2 Find the gradient of the tangent to:
a y=
x
1 ¡ 2 x
at x=1 b y=
x^3
x^2 +1
at x=¡ 1
c y=
p
x
2 x+1
at x=4 d y=
x^2
p
x^2 +5
at x=¡ 2.
3aIf y=
2
p
x
1 ¡x
, show that
dy
dx

x+1
p
x(1¡x)^2
b For what values ofxis
dy
dx
i zero ii undefined?
4aIf y=
x^2 ¡ 3 x+1
x+2
, show that
dy
dx

x^2 +4x¡ 7
(x+2)^2
b For what values ofxis
dy
dx
i zero ii undefined?
InChapter 4we saw that the simplestexponential functionshave the form f(x)=bx wherebis any
positive constant, b 6 =1.
The graphs of all members of the
exponential family f(x)=bx have the
following properties:
² pass through the point (0,1)
² asymptotic to thex-axis at one end
² lie above thex-axis for allx.

I Derivatives of exponential functions

1

y

O x

y = (0 5). x

y = (0 2). x

y=2x

y=5x

y = (1 2). x

4037 Cambridge

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Y:\HAESE\CAM4037\CamAdd_13\355CamAdd_13.cdr Friday, 4 April 2014 5:25:22 PM BRIAN