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Differential of a higher power

The differential of a higher power is

d (xn) = n xn − 1 dx

 


Explanation

You can calculate this differential with

d (xn) = (x + dx)n − xn

Use for (x + dx)n the binomium development, then you get

d (xn) =  xn  + n xn − 1 dx + ···      − xn

We only write the first two terms of the binomium, as all further terms have higher exponents of dx that are neglectable, and these will disappear, so

d (xn) = n xn − 1 dx

The formula holds for every whole, broken and negative exponent.

 


Example 0

d (x0) = 0

a = 0

 


Example 1

d (x1) = 1 dx

 

 


Example 2

d (x2) = 2x dx

 

 


Example 3

d (x3) = 3x2 dx

 


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