MULTI-COMBINATIONAL CALCULUS

INTRODUCTION TO MULTI-COMBINATIONAL CALCULUS

One of the most useful concept which I am developing is called relative comparison derivative. The concept is to help physicist, economist, scientist and engineer to compare different rate of changes of quantitative information as a fitted models for predictive purpose.

POWER RULE OF RELATIVE DERIVATIVE

For the multi-combinational derivative functions f′ (x_{o *} x_{1 *---- *} x_r-1) and
 f′ (x_1*x_2*-----* x_r), there exist unique initial derivative f′ (x_o) ,which is in relative comparison of the final derivative f ′(x_r) and is given as;

      f′(x_o) = f′ (x_r) [f′ (x_o*x_1---*x_r-1)/ f′(x_1*x_2*------*x_r)]

EXAMPLE

The gross domestic product (GDP) of Ghana, Nigeria, USA, and Indian was

N′ (t_o t₁ t₂) = t_o² t₁² t₂²  + 106 billion dollars after 2007

N′(t₁ t₂ t₃) = t²₁ t²₂ t²₃ + 106 billion dollars after 2007

(a)    At what rate was the GDP changing with respect to the multiple time t_o t₁ t₂ and t₁ t₂ t_3.

[take t_o t₁ t₂ = 8 and t₁ t₂ t₃ = 6] 

(b)   If the gross domestic product of Ghana is N (t_o) = t²_o - 5 t_o + 10 and change respectively to time to and if the gross domestic product occurs at time t_o, t_1, t₂ and t₃ for Ghana, Nigeria, USA and Indian respectively; Find the rate of change of GDP in  Indian.

(Take t_o = 3)

SOLUTION

(a)            N′ ( t_o t₁ t₂ ) = 8 ( t_o t₁ t₂ )

N′ (t₁ t₂ t₃) = 8 (t₁ t₂ t₃)

(b)   Applying power rule of relative comparison we have

N′(t_o) = N′(t₃) [N′ (t_o t₁ t₂ )/N′ (t₁ t₂ t₃ )]

But N (t_o) = 2 t_o -5

     N′(t_o) = 2(3) – 5

                = 1

1 = N′ (t₃) [(8(8))/8(6)]

1 = N′ (t₃) (1.33)

N′(t₃) =  1/1.33 = 0.75 per month after 2007