Derivative of constant and x^n.

We know the formula for finding the derivative
dy/dx = lim [f(x + δx) - f(x)]/δx
       δx→0



Constant function
For the constant function f(x) = c where c is a constant and f(x + δx) = c,
f(x + δx) - f(x) = 0
So, dy/dx = 0

Derivative of xn
For xn, f(x) = xn and f(x + δx) = (x + δx)n
f(x + δx) = (x + δx)n

By binomial expansion
= nC0xn + nC1xn-1δx + nC2xn-2(δx)2 + nC3xn-3(δx)3 + ...
= xn + (n/1!) xn-1δx + (n(n-1)/2!) xn-2(δx)2 + (n(n-1)(n-2)/3!) xn-3(δx)3 + ...

The difference
f(x + δx) - f(x) = xn + (n/1!) xn-1δx + (n(n-1)/2!) xn-2(δx)2 + (n(n-1)(n-2)/3!) xn-3(δx)3 + ... - xn
f(x + δx) - f(x) = nxn-1δx + (n(n-1)/2!) xn-2(δx)2 + (n(n-1)(n-2)/3!) xn-3(δx)3 + ...

The ratio
[f(x + δx) - f(x)]/δx = nxn-1 + (n(n-1)/2!) xn-2δx + (n(n-1)(n-2)/3!) xn-3δx2 + ...

As
dy/dx = lim [f(x + δx) - f(x)]/δx
       δx→0
 
When δx→0 all the terms containing δx becomes very small and negligible then
d(xn)/dx = nxn-1


dc/dx = 0, c is a constant
and
d(xn)/dx = nxn-1, where n is an integer.



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