Download Mathematics Drop-in Centre: Basic Differentiation and more Exercises Mathematics in PDF only on Docsity!
School of Mathematics and StatisticsThe University of New South Wales BASIC DIFFERENTIATION Mathematics Drop–in Centre
You need to know securely the derivatives of simple functions.
memorise all of the following.tions you will be using frequently, and so we recommend that youdifferentiation rules; however, you don’t want to do this for func-Some of those given below can be calculated from others by using
function
derivative
constant
x n
nx
n − 1
e x
e x
cos
(^) x
(^) sin
(^) x
sin
(^) x
cos
(^) x
tan
(^) x
sec
2 x^
ln (^) x
x 1
x 1
x 1 2
x
x
quence of the second. We have Observe that the second last entry in the above table is a conse-
x 1
=
x − 1
and so
dxd
( x 1 )
=
dxd
(^) ( x − 1 ) = (
x − 2 = (^) −
x 1 2
.
mistake of saying that the derivative of 1suggested that you memorise it, because many students make the We have, however, given this its own place in the table, and have
/x
is ln
(^) x , which is the
ers ofwrong way round. You can find the derivatives of reciprocal pow-
x
in a very similar way, for example,
d
dx
x 8 1
)
=
dxd
(^) ( x − 8 ) = (
x − 9 = (^) −
x 8 9
.
The last entry in the table can also be found from the second, √ x (^) =
(^) x 1 / 2
and so
dxd
(^) ( √ x ) =
dxd
(^) ( x 1 / 2 ) =
(^) x − 1 / 2 =
x
.
Other roots can be treated in the same way.
The
second derivative
of a function means the derivative
of the derivative. For example, the derivative of
x 7 is 7
x 6 , and so
the second derivative of
x 7 is
d 2
dx
2 (^) ( x 7 ) =
dxd
(^) (
x 6 ) = 7(
x 5 ) = 42
x 5 .
If you differentiate again, you get the
third derivative
, and so
on.
Notation
Please use notation accurately:
dxd
means “the
derivative of”, and
dxdy
means “the derivative of
(^) y ”. So “the deriva-
tive of
x 5 ” is written
dxd (^) ( x 5 ). Please do not write “
dx^ dy (^) ( x 5 )”, it is
nonsense!!
EXERCISES
Please try to complete the following exercises.
Remember that
you
(^) cannot
(^) expect to understand mathematics without doing lots
of practice!
Please do not look at the answers before trying the
please consult your tutor or the Mathematics Drop–in Centre.which you cannot find, or a question which you cannot even start,working carefully, find the mistake and fix it. If there is a mistakequestions. If you get a question wrong you should go through your
- You should be familiar with the “dash” notation for deriva-1. Write out the table of basic derivatives from memory.
tives.
For example, the second entry in the table can be
stated as “if
f (^) ( x ) =
(^) x n
then
f (^) ′ ( x ) =
nx
n − 1 ”. Write out the
whole table in this format.
- Write down the derivatives of the following functions:
x 6 ,
x 1 / 6 ,
x 6
,
tan
(^) x ,
x 1
,
ln (^) x.
- Find the derivatives of
x , 4 √ x 5 , x 3 · 14
, x − 3 · 14
, cos
(^) x .
(a) Find the second and third derivatives of ln
(^) x .
(b) Find the fourth derivative of sin
(^) x .
(c) Find the 99th derivative of
e x .
- You need to be equally comfortable with differentiation if the
variable is something other than
x
. For example, to find the
derivative with respect to
t of
t 3 we write
dt d (^) ( t 3 ) = 3
t 2 .
(a) the derivative with respect to^ Find, and write as an equation following the above example,
t of
e t ;
(b) the derivative with respect to
θ of cos
(^) θ ;
(c) the
(^) second derivative
with respect to
z
of
z 4 .
ANSWERS
x 5 , 61 (^) x − 5 / 6 , − x 6 7 (^) , sec
2 x^ , −
x 1 2 (^) ,
x 1 (^).
(^) x − 2 / 3 , 4 5 (^) x 1 / 4 , 3
· 14
x 2 · 14
, − 3 · 14
x − 4 · 14
, − (^) sin
(^) x .
(a)
x 2 (^) ,
x 2 3 (^).
(b) sin
(^) x .
(c)
e x .
(a)
dtd
(^) ( e t ) =
(^) e t ;
(b)
dθd
(^) (cos
(^) θ ) =
(^) sin
(^) θ ;
(c)
d 2
dz
2 (^) ( z 4 ) = 12
z 2 .
