Page 1 | © 2012 by Janice L. Epstein 3.2 Differentiation Formulas Page 2 | © 2012 by Janice L. Epstein g ( x) = ( x +1)(2 x - 3)
2
Differentiation Formulas (Section 3.2)
Notation
f ¢( x ) = y ¢ =
f ¢( a) =
dy
dx
dy df
d
f ( x) = Df ( x) = D f ( x)
=
=
dx dx dx
x
=
x= a
dy ù
ú
dx úû
(
F (t ) = 4 - t
x= a
The notation D and d/dx are called differentiation operators.
d
c=0
dx
d n
( x ) = nx n-1
dx
æ f ö÷¢ gf ¢ - fg ¢
çç ÷ =
çè g ÷ø
g
2
æ2ö
H ( y ) = çç ÷÷÷
èç y ø
5
( f  g )¢ = f ¢  g ¢
(cf )¢ = cf ¢
( fg )¢ = fg ¢ + gf ¢
EXAMPLE 1
Find the derivatives of the following functions
f ( x) = x + 3 x - 5 x + p - (14)
20
2
2
1- T
G (T ) =
1+ T
2
2
)
2
3.2 Differentiation Formulas Page 3 | © 2012 by Janice L. Epstein 3.2 Differentiation Formulas Page 4 | © 2012 by Janice L. Epstein 3.2 Differentiation Formulas f (u ) =
u2 - u
u
EXAMPLE 4
At what point on the curve y = x x is the tangent line parallel to
the line 3x - y + 6 = 0 ?
y = (3x + x - 7 x + 2)(-4 x + x + x - 6)
5
2
4
3
EXAMPLE 2
If f(5)=1, f’(5)=6, g(5)=-3, and g’(5)=2 find (fg)’(5)
EXAMPLE 5
Find the tangent line or lines to the parabola y = x that pass
through the point (0, -4).
2
5
4
3
2
1
-5
EXAMPLE 3
Find the equation of the tangent line to the graph of f ( x) = x + x
at the point (1, 2)
-4
-3
-2
-1
1
-1
-2
-3
-4
-5
2
3
4
5
Page 5 | © 2012 by Janice L. Epstein 3.2 Differentiation Formulas Page 6 | © 2012 by Janice L. Epstein 3.2 Differentiation Formulas EXAMPLE 6
Find where f(x) is not differentiable and graph of f(x) and f’(x)
EXAMPLE 8
2
3
2
If r (t ) = t + 2t , t + 3t is the position of a moving object at time
t, where the position is measured in feet and the time in seconds,
find the velocity and speed at time t=1.
ì-1- 2 x if x < -1
ï
ï
ï
if -1 £ x < 1
f ( x) = í x
ï
ï
if x ³ 1
ï
ï
îx
2
2
1
-2
-1
1
2
-1
-2
EXAMPLE 7
ì
x
if x £ 2
ï
ï
f
x
(
)
=
í
Given
ï
ï
îmx + b if x > 2
Find the value or values of m and b to make f(x) differentiable
everywhere.
2