Cambridge Additional Mathematics

Introduction to differential calculus (Chapter 13) 363

EXERCISE 13K

1 Find

dy

dx

for:

a y= sin(2x) b y= sinx+ cosx c y= cos(3x)¡sinx

d y= sin(x+1) e y= cos(3¡ 2 x) f y= tan(5x)

g y= sin(x 2 )¡3 cosx h y= 3 tan(¼x) i y= 4 sinx¡cos(2x)

2 Differentiate with respect tox:

a x^2 + cosx b tanx¡3 sinx c excosx d e¡xsinx

e ln(sinx) f e^2 xtanx g sin(3x) h cos(x 2 )

i 3 tan(2x) j xcosx k

sinx

x

l xtanx

3 Differentiate with respect tox:

a sin(x^2 ) b cos(

p

x) c

p

cosx d sin^2 x

e cos^3 x f cosxsin(2x) g cos(cosx) h cos^3 (4x)

i

1

sinx

j

1

cos(2x)

k

2

sin^2 (2x)

l

8

tan^3 (x 2 )

4 Find the gradient of the tangent to:

a f(x) = sin^3 x at the point where x=^23 ¼

b f(x) = cosxsinx at the point where x=¼ 4.

Given a function f(x), the derivative f^0 (x) is known as thefirst derivative.

Thesecond derivativeof f(x) is the derivative of f^0 (x),orthe derivative of the first derivative.

We use f^00 (x) or y^00 or

d^2 y

dx^2

to represent the second derivative.

f^00 (x) reads “fdouble dashedx”.

d^2 y

dx^2

=

d

dx

³dy

dx

́

reads “dee twoyby deexsquared”.

Example 18 Self Tutor

Find f^00 (x) given that f(x)=x^3 ¡^3

x

Now f(x)=x^3 ¡ 3 x¡^1

) f^0 (x)=3x^2 +3x¡^2

) f^00 (x)=6x¡ 6 x¡^3

=6x¡

6

x^3

L Second derivatives

4037 Cambridge

cyan magenta yellow black Additional Mathematics

(^05255075950525507595)
100 100
(^05255075950525507595)
100 100
Y:\HAESE\CAM4037\CamAdd_13\363CamAdd_13.cdr Tuesday, 7 January 2014 9:55:13 AM BRIAN