Ta b l e 8. 1

Functionf(x) Derivativef′(x)

c=constant 0

xn nxn−^1 n = 0

xn+^1 /(n+ 1 )xn(n  =− 1 )

ln|x|=loge|x| 1 /x

sinx cosx

cosx −sinx

tanx sec^2 x

ex ex

ln[u(x)]

u′(x)

u(x)

eu(x) u′(x)eu(x)

Integral

∫

g(x)dx Functiong(x)

Solution to review question 8.1.3

(i)y=49 is a constant, so its derivative is 0. Think of it as a quantity

whose rate of change or gradient is zero – its graph is always ‘flat’.

(ii) y=x^4 ,sousing

d(xn)

dx

=nxn−^1 withn=4gives

dy

dx

= 4 x^3

(iii) y=

√

x=x

1

(^2) so
dy
dx

1
2
x−
1
(^2) =
1
2
√
x
(iv)y=
1
x^2
=x−^2 so
dy
dx
=− 2 x−^3 =−
2
x^3
(v)y=sinx,so
dy
dx
=cosxdirectly from Table 8.1. Note thatxmust
be in radians here.
(vi)y=exso
dy
dx
=ex again from the table.ex is unique in being
its own derivative and this is partly responsible for its immense
importance.
(vii) Fory=lnx,
dy
dx

1
x
from Table 8.1.
(viii) y= 2 x
This question is a bit naughty, since itdoes not come from our list
of standard integrals, but beginners sometimes make the error of
thinking it does and write
‘d
dx
( 2 x)=x 2 x−^1 ’