Post on 17-Jan-2016
Calculus and Analytical Geometry
Lecture # 8
MTH 104
Techniques of differentiation
1. Constant Function Rule:The derivative of a constant function is zero. y = f(x) = cwhere c is a constant
.0
dxdc
dxxdf
dxdy
Examples
,0)1(
dxd ,0
)5( dxd .0
)2( dx
d
Techniques of differentiation
nxxfy )(
xxxdxd
22 122
1 nn nxxdxd
2. Power Rule:Let , where the dependant
variable x is raised to a constant value, the power n, then
5xdxd
45x
Examples
Techniques of differentiation
2
1
xdxd
xxx
21
21
21
2
11
2
1
7xdxd 817 77 xx
78 8xxdxd
3. Constant Multiplied by a Function Rule:Let y be equal to the product of a constant c and some function f(x), such that y = cf(x) then
Techniques of differentiation
dxxdf
cdxxcfd
dxdy )())((
dxxd )4( 3
dxxd )(4 3
Examples
213 1234 xx
Techniques of differentiation
1112
1212
x
xdxd
xdxd
1
2
1
x
xdxd
xdxd
xdxd
Techniques of differentiation
xgdxd
xfdxd
xgxfdxd
dxdy
962 xxdxd 962 x
dxd
xdxd
4. Sum (Difference) Rule:Let y be the sum (difference) of two functions (differentiable) f(x) and g(x).
y = f(x) + g(x),
then
Examples
)9(62 195 xx105 912 xx
Techniques of differentiation
x
x
x
xdxd
dxd
xdxd
1
21
-2
dxd
2-0
2121
2
1
2
1
Techniques of differentiation.9523 38 xxxy
9523 38 xxxdxd
dxdy
9523 38
dxd
xdxd
xdxd
xdxd
Example Find dy/dx if
solution
0)1(53283 27 xx
5624 27 xx
Techniques of differentiation
433 xxy
33
43
2
3
x
xxdxd
dxdy
Example At what points, if any does the graph of
have a horizontal tangent line? solution
1
1
01
033 0
2
2
2
x
x
x
xdxdy
Slope of horizontal line is zero that is dy/dx=0
Techniques of differentiation
4. Product Rule:Let y = f(x).g(x), where f(x) and g(x) are two
differentiable functions of the variable x. Then
xfdxd
xgxgdxd
xfdxdy
xgxfdd
dxdy
Techniques of differentiation
xxxy 32 714
xxxdxd
dxdy 32 714
147714 2332 xdxd
xxxxdxd
x
Example Find dy/dx, if
solution
xxxxx 8712114 322
19140
85612148424
24224
xx
xxxxx
Techniques of differentiation
5. Quotient Rule:Let y = f(x)/g(x), where f(x) and g(x) are two differentiable functions of the variable x and g(x) ≠ 0. Then
2
xg
xgdxd
xfxfdxd
xg
dxdy
xgxf
dxd
dxdy
Techniques of differentiation
514
2
xx
y
514
2xx
dxd
dxdy
22
22
5
5)14(145
x
xdxd
xxdxd
x
Example Find dy/dx if
solution Derivative of numerator
Derivative of denominator
Techniques of differentiation
22
2
52)14(45
xxxx
22
22
528204
xxxx
22
2
52024
xxx
Higher order derivatives
nfffff 4,,,If y=f(x) then
xfdxd
dxyd
y
xfdxd
xfdxd
dxd
dxyd
y
xfdxd
dxdy
y
3
3
3
3
2
2
2
2
Higher order derivatives
A general nth order derivative
xf
dxyd
xfxfdxd
dxyd
n
n
n
n
n
n
n
n
and
Example
constants. are
,, where if Find 24 cbacbxaxyy
cbxaxdxd
dxdy
y 24
cdxd
bxdxd
axdxd 24
Solution
bxax 24 3 bxax
dxd
dxyd
y 24 3
2
2
First Orderderivat
ive
Second orderderivative
)2()4(2
2
3
2
2
2
2
bxdxd
axdxd
dxyd
y
bax 212 2
ax
baxdxd
dxyd
y
12
212 2
3
3
3
3
Third order derivative
Example Find 25
1x
2
2
46 where xxydx
yd
xx
xx
xxdxd
dxdy
830
2456
46
4
4
25
xxdxd
dxyd
830 4
2
2
Solution
8120 3 x
1128)1(1201
2
2
x
dx
yd
Derivative of trigonometric functions
xxdxd
cossin .1
xcoxdxd
sin .2
xxdxd
2sectan .3
xxxdxd
tansecsec .5
xxxdxd
cotcsccsc .4
xxdxd
2csccot .6
Example . find cossin2
2
dxyd
xxy
xxdxd
dxdy
cossin
xdxd
xxdxd
x sincoscossin
Solution
)(coscos)sin(sin xxxx
xx 22 cossin
xdxd
xdxd
dxyd
22
2
2
cossin
xx
xxxx
cossin4
cossin2cossin2
Example
xyy
xxy
cos2
osolution t a is sin that Show
1 cos2 xyy
xxxy r.t w.sin times twoatingDifferenti
solution
xxx
xxdxd
dxdy
y
sincos
sin
xdxd
xxdxd
xxxdxd
y sincossincos
xxx
xxxxy
cos2sin
coscossin
yy and
xxxxxx cos2sincos2sin
Substituting the valuse of into (1)
xx cos2cos2 L.H.S=R.H.S
Example Given that xxxf tansin)( 2 24
fshow that
xxxf tansin)( 2
xxxf tansin)( 2xx 22 secsin )(xf ))(cos(sin2 tan xxx
xx
22
cos
1sin
x2tan x2sin2
4f
4tan2
4
sin2 2
12
2
12
2
1
))(cos(sin2
cos
sin xx
x
x