 # RULES OF DIFFERENTIATION (CONT)

Examples

Find the derivative of each of the following:

HIGHER DERIVATIVES OF THE SECOND  AND THIRD ORDER . DIFFERENTIATION OF IMPLICIT FUNCTIONS

HIGHER  DERIVATIVES

EVALUATION

Find the second and third derivatives of  (1) cos 6x  ( 2)  4x5 -5x

Implicit Differentiation

So far, we have treated relations. Of the form y = f (x). Examples of such relations are y = 3x2 – 2x + 1,  y = 1 +

In any of these relations, y is said to be expressed explicitly in terms of x. The derivative of y with respect to x can be found from the rules of differentiation which have been discussed in the previous units.

Sometimes, the relationship between y and x may not be expressed explicitly.

For example, consider x2y + xy3 + 3x = 0. Here, the relation between y and x is not expressed explicitly. The relationship between y and x is said to be implicit.

In differentiating x2y+xy3+3x=0, y is treated as if it is a function of x and the rules of differentiation are applied in the appropriate manner. The process of differentiating implicit function is called implicit differentiation.

EVALUATION

Differentiate the followings ;

(i) y= (3x+4) (6x-8)

(ii)  y = 6x+7/2x-3

GENERAL  EVALUATION

1) Differentiate  y = ( 7x4 – 6 )5

2) Differentiate  y = ( 2x + 5) ( 6x – 8)

3) Find the derivative of  y  =  3x2 – 5/x + 3

4) Find the derivative of  y  = 8/ ( 9 – x5)4

5) Find the derivative of  y = 2x4 -5x3 -+ 6

6) If  x3– y2 + 6xy = 0 find dy/dx

7) Find  d3y/dx3  given that   y = 8x5 – 3x4 + 9x3 -7x2 +6x+4

New Further MathsProject  2 page 121 – 126

WEEKEND ASSIGNMENT

1) If  y = 3x4 -7x + 5  find dy/dx   a) 12x3  b) 12x3 – 7  c) 12x3 + 5  d) 12x3 + 12

2) Find the second derivative of cos 5x   a) 5sin5x  b) -25cos5x  c) 25cos5x  d) -25sin5x

3) 2) If x2y + 4xy =1 find  dy/dx  a)  4+2xy/x2  b) 4-2xy/x2   c) -4-2xy/x2  d) -4+2xy/x

4) Given that  y = x2 + 3x + 2,   find dy/dx   at x = 2  a)  6  b) 4  c) 7  d) 5

5) Given that   y = ( 2x + 3)4 find dy/dx  a) 18(2x + 3)3  b) 4(2x + 3)4  c) 8(2x + 3)3  d) 2( 2x+3)3

THEORY

1) Differentiate  y = (2x2 -3)3/x

2) Differentiate  y = (2x+ 3)3 (4x2 -1)2