Derivatives of functions

2. • Higher Order derivatives
• Polar Curves

4. The Product Rule can be extended to cover
products
involving more than two factors. For
example, if f, g, and h
are differentiable functions of x, then
So, the derivative of y = x2 sin x cos x is

5. Find the derivative of
Solution:

7. Find the derivative of
Solution:

9. The summary below shows that much of the work in obtaining a
simplified form of a derivative occurs after differentiating. Note
that two characteristics of a simplified form are the absence of
negative exponents and the combining of like terms.

10. Higher-order derivatives are denoted as follows.

11. To find the acceleration, differentiate the position function
twice.
s(t) = –0.81t2 + 2 Position function
s'(t) = –1.62t Velocity function
s"(t) = –1.62 Acceleration function
So, the acceleration due to gravity on the moon is –1.62
meters per second per second.
Example

12. 12
The center of the graph is
called the pole.
Angles are measured from
the positive x axis.
Points are
represented by a
radius and an angle
(r, )
radius angle
To plot the point






4
,5

First find the angle
Then move out along
the terminal side 5

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All rights reserved. 13
The polar coordinate system is formed by fixing a point, O,
which is the pole (or origin).
 = directed angle Polar
axis
O
Pole (Origin)
The polar axis is the ray constructed from O.
Each point P in the plane can be assigned polar coordinates (r, ).
P = (r, )
r is the directed distance from O to P.
 is the directed angle (counterclockwise) from the polar axis
to OP.

14. 14
A negative angle would be measured clockwise like usual.
To plot a point with
a negative radius,
find the terminal
side of the angle
but then measure
from the pole in
the negative
direction of the
terminal side.







4
3
,3








3
2
,4


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All rights reserved. 15
(r, )
(x, y)
Pole
x
y
(Origin)
y
r
x
The relationship between rectangular and polar
coordinates is as follows.
The point (x, y) lies on a
circle of radius r, therefore,
r2 = x2 + y2.
tan
y
x
 
cos x
r
 
sin
y
r
 
Definitions of
trigonometric functions

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All rights reserved. 16
Coordinate Conversion
cosx r cos x
r
 
siny r sin
y
r
 
2 2 2
r x y tan
y
x
  (Pythagorean Identity)
Example:
Convert the point into rectangular coordinates. 4,
3

   1cos co
3
24 s 4
2
x r   
  3sin sin 4 2
3 2
4 3y r 
 
      
 

   , 2, 2 3x y  

17. 17
The length of an arc (in a circle) is given by r.  when  is
given in radians.
Area Inside a Polar Graph:
For a very small , the curve could be approximated by a
straight line and the area could be found using the triangle
formula: 1
2
A bh
r dr
  21 1
2 2
dA rd r r d   

18. 18
We can use this to find the area inside a polar graph.
21
2
dA r d
21
2
dA r d
21
2
A r d


 

19. 19
Example: Find the area enclosed by:  2 1 cosr  
-2
-1
0
1
2
1 2 3 4
2
2
0
1
2
r d
 




 
2 2
0
1
4 1 cos
2
d

   
 
2
2
0
2 1 2cos cos d

    
2
0
1 cos2
2 4cos 2
2
d
 
 

   

20. 20
2
0
1 cos2
2 4cos 2
2
d
 
 

   
2
0
3 4cos cos2 d

    
2
0
1
3 4sin sin 2
2

    
6 0 
6

21. 21
Notes:
To find the area between curves, subtract:
2 21
2
A R r d


 
Just like finding the areas between Cartesian curves,
establish limits of integration where the curves cross.

22. 22
To find the length of a curve:
Remember: 2 2
ds dx dy 
Again, for polar graphs: cos sinx r y r  
If we find derivatives and plug them into the formula,
we (eventually) get:
2
2 dr
ds r d
d


 
   
 
So: 2
2
Length
dr
r d
d




 
   
 


23. 23
2
2
Length
dr
r d
d




 
   
 

There is also a surface area equation similar to the
others we are already familiar with:
2
2
S 2
dr
y r d
d


 

 
   
 

When rotated about the x-axis:
2
2
S 2 sin
dr
r r d
d


  

 
   
 
