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