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Lesson 11

Higher Order and

Implicit Differentiation

OBJECTIVES:

• to define higher derivatives;
• to apply the knowledge of higher derivatives

and implicit differentiation in proving relations;

• to find the higher derivative of algebraic

functions; and

• to determine the derivative of algebraic

functions implicitly under the specified conditions.

• Other common notations for higher derivatives are the following:

first derivative:

, ', '( ), f (x), D f (x) dx

d y f x dx

dy x

second derivative:

, '', ''( ), 2 ( ),^2 ( )

2 2

2 f x D f x dx

d y f x dx

d y x

nth derivative:

, , ( ), f (x),D f (x ) dx

d y f x dx

d y n nx

n n n n

n

The symbols , dx

dy (^) , 2

2 dx

d y n

n dx

d y are called Leibniz notations.

y =− 6 x^5 + 5 x^4 − 2 x^3 + 3 x^2 + 10 x− 5

1. Find all the derivatives of the function.

y' = − 30 x^4 + 20 x^3 − 6 x^2 + 6 x+ 10

EXAMPLE

S:

SOLUTIO

N:

y (^6 )^ = 0

y" = − 120 x^3 + 60 x^2 − 12 x+ 6 y ''^ ' = − 360 x^2 + 120 x− 12 y (^4 )^ = − 720 x+ 120 y (^5 )^ = − 720

1 x

of y x dx

3. Find d y 2

2 −

=

( )^2 ( )^2 ( 1 x)^2

1 1 x

1 x x 1 x

1 x 1 x 1 dx

dy −

= −

= − + −

= − − −

( ) ( ) [ ( )( )] ( )

( ) ( )

(^2) ( ) 3

2

4 4

2 2

2

1 x

2 dx

d y

1 x

21 x 1 x

1 x 0 121 x 1 dx

d y

−

=

−

= − −

= − − − −

SOLUTIO

N:

An implicit function is a function in which the dependent variable has not been given "explicitly" in terms of the independent variable. To give a function f explicitly is to provide a prescription for determining the output value of the function y in terms of the input value x : y = f ( x ). By contrast, the function is implicit if the value of y is obtained from x by solving an equation of the form: R ( x , y ) = 0.

Steps in Implicit Differentiation

1. Differentiate both sides of the equation with

respect to x.

2. Collect all the terms with on one side of the equation.
3. Factor out.
4. Solve for.

dx

dy

dx

dy

dx

dy

1. Find y'and y'' of x^2 − 4 y^2 = 4 by implicit differentiation.

4 y y' x

8 y y'^2 x

Solve for y' ,

8 yy' 2 x

2 x 8 yy' 0

eachtermimplicitly,

Takethederivativeof

x 2 4 y^24

=

=

=

− =

− =

4 y thenreplace y' x y

y-xy' 4 y''^1

Find y'' (^2)  =

  = 

3 3

2 2

3

2 2 3

2 2

2

2 2

2

4 y

1 16 y y"^4

but x 4 y 4

16 y

x 4 y 4 y

4 y x 4 y"^1

y

4 y

4 y x

4

1 y

4 y y x x 4 y"^1

= − =^ −

− =

= • − = − −

− = •



  −  = •

EXAMPLE

Solutio

n:

( )

y

x dy x dx dy ydx

x 1 ydydx 0

Now find dydx in xy 1.

=− ⇒ =^ −

• =

=

( ( ))

2 2

2

2 2

2

y

y dydx x dy

d x

y

y( dydx) x 1 dy

d x

Differentiate again wrt y.

^ + 

 

 −

− − −

y" x" 4 since xy 1

xy

4 y

2 x x y" x"^2 y

Therefore, 2 2 ⋅ = =

⋅ = ⋅ =

ddyx (^2) yx this is x "

y

x x y

y yx x dy

d x

y

x dy then substitute dx

2 2

2

2 2 2

2

= ⇒

+ = +    (^) −− =

=−