It is all about calculus. It will help you to be more familiar in derivatives of a function. Study it and you will get some high grades on your calculus subject. Hope you will be able to master it and teach it to your classmates. Thank you and study harder to get high scores and grades.. Ajajajajajajajajajajajajajajajajajajajajajajajajajajajajajajajajajajajajajajajajajajajajajajajajajajajajajajajajajajajaja

1. 3.1 Derivatives
Great Sand Dunes National Monument, Colorado
Greg Kelly, Hanford High School, Richland, Washington
Photo by Vickie Kelly, 2003

2.    
0
lim
h
f a h f a
h

 
is called the derivative of at .
f a
We write:  
   
0
lim
h
f x h f x
f x
h

 
 
“The derivative of f with respect to x is …”
There are many ways to write the derivative of  
y f x



3.  
f x
 “f prime x” or “the derivative of f with respect
to x”
y “y prime”
dy
dx
“dee why dee ecks” or “the derivative of y with
respect to x”
df
dx
“dee eff dee ecks” or “the derivative of f with
respect to x”
 
d
f x
dx
“dee dee ecks uv eff uv ecks” or “the derivative
of f of x”
( of of )
d dx f x


4. dx does not mean d times x !
dy does not mean d times y !


5. dy
dx does not mean !
dy dx

(except when it is convenient to think of it as division.)
df
dx
does not mean !
df dx

(except when it is convenient to think of it as division.)


6. (except when it is convenient to treat it that way.)
 
d
f x
dx
does not mean times !
d
dx
 
f x


7. In the future, all will become clear. 

8. 0
1
2
3
4
1 2 3 4 5 6 7 8 9
 
y f x

-2
-1
0
1
2
3
1 2 3 4 5 6 7 8 9
 
y f x


The derivative
is the slope of
the original
function.
The derivative is defined at the end points
of a function on a closed interval.


9. -3
-2
-1
0
1
2
3
4
5
6
-3 -2 -1 1 2 3
x
2
3
y x
 
   
2 2
0
3 3
lim
h
x h x
y
h

   
 
2 2 2
0
2
lim
h
x xh h x
y
h

  
 
2
y x
 
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
-3 -2 -1 1 2 3
x
0
lim2
h
y x h

  
0


10. A function is differentiable if it has a
derivative everywhere in its domain. It
must be continuous and smooth.
Functions on closed intervals must have
one-sided derivatives defined at the end
points.
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