Math 151
Section 3.2
Differentiation Formulas
Constant Rule If f is a constant function, f ( x) = c , then f !( x ) = 0 .
Power Rule If f ( x ) = x n , where n is any real number, then f !( x ) = nx n"1 .
Product Rule If y = f ( x) g ( x) and f !( x) and g !( x) both exist, then
y ! = f !( x) g ( x) + f ( x) g !( x)
Quotient Rule If y =
y! =
f ( x)
g ( x)
and f !( x) and g !( x) both exist, then
g ( x) f !( x) " f ( x) g !( x)
( g ( x))
2
Lots and Lots of Examples: Find the following derivatives.
A. y = 5
y = !2
B. y = x10
C. y = 3x 7
D. h( x) = 3+ x 5
y = sin18°
Math 151
E. f ( x) = 5x 3 + 2x !1
F. y = 3 x 8 + x
G. y = 3x!2 +
H. g (t ) =
I. y =
1
+ 3!1
3
2x
t 3 + 5t 2 + 7
t
x 4 !1
x2 x
Math 151
J. y = ( x 6 + 7)( x 2 + x)
K. y = ( x 4 + 3x 2 + 2)( x 2 ! x + 4)
L. f ( x) =
M. y =
1! x 2
1+ x 2
5
x +2
6
Math 151
#%1! 2x
%%
N. g ( x) = %
$x 2
%%
%%& x
if x < !1
if !1" x "1
if x >1
#4x 2 + 2x +1 if x <1
%
O. k ( x) = %
$
%
if x "1
%
&10x ! 3
Example: Assume that r(t) is a position function for an object. Find the velocity vector(s) at the
point (3, 0) for r (t ) = t 2 ! 6t + 8,t 4 ! 26t 2 + 25 .
Example: For what value(s) of x is the tangent line of f ( x) = x 3 !5x 2 + 6x ! 30 parallel to
y = 6x + 5 ?
Math 151
Example: The functions f and g and the corresponding derivatives have values as shown in the table.
x
f ( x)
f !( x)
g ( x)
g !( x)
0
1
3
5
−3
1
2
9
7
11
2
0
2
10
−5
3
4
8
−1
−4
Use the table to evaluate the following:
A. h!(3) for h( x) = ( x 3 + 2) g ( x)
B.
d !## x 3 $&&
&
#
dx ##" f ( x)&&%
x=1