RULES OF DIFFERENTIATION (CONT)

Back to: Further Mathematics SS2

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

Reading Assignment

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

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