Introduction to differential calculus (Chapter 13) 347

Example 8 Self Tutor

Find the gradient function for each of the following:

a f(x)=3

p

x+

2

x

b g(x)=x^2 ¡

4

p

x

a f(x)=3

p

x+

2

x

=3x

1

(^2) +2x¡^1
) f^0 (x)=3(^12 x
¡^12
)+2(¡ 1 x¡^2 )
=^32 x
¡^12
¡ 2 x¡^2

3
2
p
x
¡
2
x^2
b g(x)=x^2 ¡
4
p
x
=x^2 ¡ 4 x
¡^12
) g^0 (x)=2x¡4(¡^12 x
¡^32
)
=2x+2x
¡^32
=2x+
2
x
p
x

EXERCISE 13E

1 Find f^0 (x) given that f(x) is:

a x^3 b 2 x^3 c 7 x^2 d 6

p

x

e 33

p

x f x^2 +x g 4 ¡ 2 x^2 h x^2 +3x¡ 5

i^12 x^4 ¡ 6 x^2 j^3 x¡^6

x

k^2 x¡^3

x^2

l x

(^3) +5
x
m
x^3 +x¡ 3
x
n
1
p
x
o (2x¡1)^2 p (x+2)^3
2 Find
dy
dx
for:
a y=2: 5 x^3 ¡ 1 : 4 x^2 ¡ 1 : 3 b y=¼x^2 c y=
1
5 x^2
d y= 100x e y= 10(x+1) f y=4¼x^3
3 Differentiate with respect tox:
a 6 x+2 b x
p
x c (5¡x)^2 d
6 x^2 ¡ 9 x^4
3 x
e (x+ 1)(x¡2) f
1
x^2
+6
p
x g 4 x¡
1
4 x
h x(x+ 1)(2x¡5)
4 Find the gradient of the tangent to:
a y=x^2 at x=2 b y=
8
x^2
at the point (9, 818 )
c y=2x^2 ¡ 3 x+7 at x=¡ 1 d y=
2 x^2 ¡ 5
x
at the point (2,^32 )
e y=
x^2 ¡ 4
x^2
at the point (4,^34 ) f y=
x^3 ¡ 4 x¡ 8
x^2
at x=¡ 1.
5 Suppose f(x)=x^2 +(b+1)x+2c, f(2) = 4, and f^0 (¡1) = 2.
Find the constantsbandc.
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Y:\HAESE\CAM4037\CamAdd_13\347CamAdd_13.cdr Tuesday, 7 January 2014 2:37:35 PM BRIAN