## Derivative Of A Constant

In this page derivative of a constant we are going to see the formula derivation for constant.

The derivative of a constant function is zero.

That is d(c)/dx = 0   where c is constant

Now we ahve to put x as x + Δx

let f(x)  = c      then f (x + Δx) = c

d (f(x))/dx = lim      [ f (x + Δx) - f(x)] /Δx

Δx --> 0

d (c)/dx =   lim       [ c - c] /Δx

Δx --> 0

d (c)/dx =   lim       0 /Δx

Δx --> 0

=  0

The derivative x is nxⁿ-¹,where n is a rational number.

d (xⁿ)/dx = n x ⁿ-¹

let f(x)  = xⁿ   then f(x + Δx) = (x + Δx)ⁿ

d (f(x))/dx = lim    [ f (x + Δx) - f(x)] /Δx

Δx --> 0

d (xⁿ)/dx = lim    [ f (x + Δx)ⁿ - xⁿ] /Δx

Δx --> 0

d (xⁿ)/dx = lim    xⁿ [ (1 + (Δx/x))ⁿ - xⁿ] /Δx

Δx --> 0

d (xⁿ)/dx = lim    xⁿ [ (1 + (Δx/x))ⁿ - 1] /Δx

Δx --> 0

d (xⁿ)/dx = lim    xⁿ  x-¹ [ (1 + (Δx/x))ⁿ - 1] /(Δx/x)

Δx --> 0

d (xⁿ)/dx = lim    xⁿ-¹ [ (1 + (Δx/x))ⁿ - 1] /(Δx/x)

Δx --> 0

Now we have to put y = 1+(Δx/x) as Δx --> 0 ,y --> 1

d (xⁿ)/dx =  xⁿ-¹   lim      (yⁿ - 1) /(y-1)

y --> 1

=  n xⁿ-¹