ADVANCED MATHEMATICS FORM 5 – DIFFERENTIATION

DERIVATIVES

Slope of a curve

A curve has different slopes at each point. Let A, B, and C be different points of  a curve f (x)

Where ðx is the small increase in x

            ðy is the small increase in y

The slope of chord AC =                    

If C moves right up to A the chord AC becomes the tangent to the curve at A and the slope at A is the limiting value of   

                                                               

Therefore

       =

The gradient at A is

=

or

This is known as differentiating by first principle

From the first principle

i)                   f(x)= x

ii)                f(x)= x²

= 

∴

iii)           f(x) =x³

iv)    f(x)= xⁿ

By binomial series

In general

If

 

Example

Differentiate the following with respect to x

i)y = x²+3x

Solution

y =x²+3x

ii)    2x⁴+5

iii)    

=

Differentiation of products functions [ product rule]

Let y =uv

Where u and v are functions of x

If x   →   x+ðx

u     →    u +ðu

v     →   v+ðv

y     →    ðy +y

y= uv ……i)

Therefore

y+ðy = [ u+ðu][v+ðv]

y+ðy = uv +uðv+vðu +ðuðv….ii

Subtract (i) from (ii)

δy  =uðv +vðu +ðuðv

Therefore

  

Therefore

Therefore
If y= uv

    It is the product rule

Examples

Differentiate the following with respect to x

i)    y = [ x²+3x] [4x+3]

ii)    y = [  +2] [x²+2]

Solution

Y = [x²+3x] [4x+3]

Let u = x²+3x

 = 2x+3

V = 4x+3

Therefore

=4x²+12x+8x²+12x+6x+9

=12x²+30x+9

ii)Let u =  +2       →

v = x²+2       =2x

Therefore