f x h f x dy f x y lim - - PowerPoint PPT Presentation

* Definition of Derivative:

The first derivative of the function with respect to the variable x is the function whose value at x is:

 

y f x 

f 

     

lim

h

f x h f x dy f x y dx h



      

provided the limit exists.

Important theorems:

• 1. If y = f(x) is differentiable at x = a, then y = f(x) is

continuous at a . The inverse is not always true.

• 2. If the function y = f(x) is discontinuous at the point

x = a, then it is not differentiable at this point.

Geometric Interpretation of Derivative:

* The slope of tangent line to the graph of the function f(x) at (a,f(a)) is the derivative of f(x) at x = a.

a

  

h

y x

.

 

x f y 

h a

P

   

a f h a f  

  

h

.

      

lim

h

f a h f a f a h



   

= Slope of tangent at P

* we can write the equation of the tangent line to the curve at the point (a, f(a)):

     

y f a f a x a    

Example Find an equation of the tangent line to the curve

9 x 8 x y

2

  

at the point (3,- 6). Solution

8 x 2 y   

   

3 2 3 8 2 y     

   

3 x 2 6 y     

2 y x  

Rules of Differentiation:

 

 

 

 

d d c f x c f x dx dx 

 

 

2

3 3 2 6 d x x x dx  

 

f g f g      

 

3 2

6 5 3 6 d x x x dx    

 

f g g f g f     

 



 

   

3 2 3

2 3 2 2 3 3 1 2 2 d x x x dx x x x x             

 

2

f f g g f g g           

   



  

4 3 4 2 4

1 1 4 1 1 x x x x d x dx x x x x              

Tables of Differentiation

Table (1)

 

f x

 

f x 

n

x

1 n

nx



1 x

lnx

x

e

x

e

x

a

ln

x

a a

(const k ant)

k x

k

x

1 2 x

Table (2)

 

f x

 

f x  cosx

sinx

2

sec x

tanx

2

csc x

cotx

sec tan x x secx

sinx

cosx Table (3)

Trigonometric Functions Hyperbolic Functions

csc cot x x cscx

 

f x

 

f x 

coshx sinhx

2

sech x

tanhx

2

csch x

cothx

sec t nh a x h x sechx sinhx coshx

csc c th

• x

h x cschx

Example

Differentiate the functions:

3

) sin a y x x  

) cosh b y x x 

3

x x c y x   cos ) sin

2

3 cos y x x   

 

1 sinh cosh 2 y x x x x   

  



1 2 3 2 2

3 2 x x x x x x y x           

/ /

sin sin cos cos sin

Example:

Obtain the derivative of tan x from sin x and cos x.

Solution:

 

tan d x dx

sin cos d x dx x       

 

2

cos cos sin sin cos x x x x x   

2 2 2

cos sin cos x x x  

2 2

1 sec cos x x  

The Derivative of a Composite Function

 

 

d f g h x dx     

 

g h x  

x  

 

h x  

 

f g h x    

Differentiate the functions:

 

2

i ) s n a y x 

 

2

cos x y  

3

y x b x         cos n ) si

3 '

sin sin x x y   

3

1 2 sin x x  

 

2

3 cos x x  

 

x 2 

Examples

Examples

 

5

1 tan cos h 1 y x       

 

4 5

1 5 y hx



     

/

tan sec

 

2

hx sec sec

hx x sec tanh

 

1 5

tan sechx     

Example

 

 

 

3

2

ln co 2 t

x

y e x 

 

 

3

2

ln cot

x

e x 

3

x

y e   

 

2 x  cot

 

 

2

x   ln cot

3

x

e

 

2

1 x cot

[

]

 

2 x

 csc

[

]

2

3x 