Derivatives of Bases Other than e

Derivatives:

• f(x) = a^x

       = e ^{(ln a) x} = e ^{x ln a}   so   f '(x) = (ln a) e^{x ln a} = (ln a) a^x
• f(x) = log_ax

       =  ln x so   f '(x) =   1      1     =      1
           ln a                    ln a   x         x ln a

Figuring Amount in Savings Account:

Find the amount of money in a savings account after n years.

• A₁ = A₀ +(r/4) A₀  =  A₀ (1 + r/4 )

 
• A₂ =   A₀ (1 + r/4 ) + (r/4) A₀(1 +  r/4  ) =  A₀ (1 +  r/4  )(1 +  r/4  ) + A₀(1 + r/4)²
• Similarly,   A₃ = A₀(1 + r/4)³
• and  A_n = A₀(1 + r/4)ⁿ

Formula for t years:

Derive the formula for compounding continuously:

• A_t = A₀(1 + r/n)^nt

     = A₀ [(1 + r/n) ⁿ] ^t

• Let f(n) = (1 + r/n)ⁿ

 ln f(n) = n ln (1 + r/n)
• Let r/n = h

         r = nh
  so   n = r/h
As n®oo, h®0
• Since  ln f(n) = n ln (1 + r/n)   (from above)

          ln f(n) = r/h ln (1 + h)   (substitute expressions in)
                     = r ln (1 + h)       (rewrite)
                                h
• lim       ln (1 + h)

h®0          h

=   lim          1      (using L'Hopital's Rule)
    h®0    1 + h

=  1

• Since  ln f(n) = n ln (1 + r/n)   (from above)

and           = r ln (1 + h)       (from above)
                                h

lim   ln f(n)
h®0

=  lim    r ln (1 + h)
    h®0         h

=  r(1)

=  r

• Thus,   f(n) ® e ^r  as  n®oo

 
• so   lim       A_t  =  lim       A₀ [(1 + r/n) ⁿ] ^t

      n®oo            n®oo

   =   A₀ e ^rt