Transcript POWER SERIES - MATHCHICK.NET

BASIC DIFFERENTIATION
RULES AND RATES OF
CHANGE
Section 2.2
When you are done with your
homework, you should be able
to…
• Find the derivative of a function using the
Constant Rule
• Find the derivative of a function using the Power
Rule
• Find the derivative of a function using the
Constant Multiple Rule
• Find the derivative of a function using the Sum
and Difference Rules
• Find the derivatives of the sine function and the
cosine function
• Use derivatives to find rates of change
THE CONSTANT RULE
The derivative of a constant function is
zero. That is, if c is a real number,
then
d
c  0
dx
Find the slope of f  x   3
A. -3
B. 0
C. undefined
Evaluate f '  x  , if f  x   3,
using the definition of the derivative
A.
B.
C.
D.
-3
0
undefined
?
THE POWER RULE
If n is a rational number, then the function
d n
n
f  x   x is differentiable and
x  nx n 1.
dx
 
For f to be differentiable at x  0, n must
n 1
x
be a number such that
is defined on an
interval containing 0.
d 5
Evaluate
 x 
dx
4
x
A.
4
5x
B.
5
C. 5x
5
4x
D.
d  
Evaluate
x
dx  
A. 2x
B. 1
2 x
1
3
C. x 2
2
 12
D. Both A and B
THE CONSTANT
MULTIPLE RULE
If f is a differentiable function and c is a
real number, then cf is also
differentiable and
d
cf  x    cf   x 
dx
d 6
Evaluate
2

dx  x 
1
A. 3x
3
B. 3x
3
C. 6x
3
D. 12x
Evaluate the derivative of
5
y  2 x at x  3.
A.
B.
C.
D.
81
-162
-810
-486
THE SUM AND DIFFERENCE
RULES
• The sum or difference of two
differentiable functions f and g is
itself differentiable.
– The derivative of the sum or difference
of functions is the sum or difference of
the derivatives of f and g.
d
 f  x   g  x   
dx
d
 f  x   g  x   
dx
f  x  g x
f  x  g x
DERIVATIVES OF SINE AND
COSINE FUNCTIONS

 sin x   cos x
x

 cos x    sin x
x
Evaluate the slope of the graph f

at x  .
6
A.
B.
C.
D.
1/2
3 2
-1/2
2 2
 x   cos x
RATES OF CHANGE
• Velocity
– Average Velocity
change in distance s

change in time
t
– Instantaneous Velocity
s  t  t   s  t 
v  t   lim
 s  t 
t 0
t