In this section, we will obtain the derivatives of logarithmic and exponential functions. We begin with the derivative of logax, where a and x are positive.

Derivative of log_ax.

Let

f (x) = log_ax

Thens  f ' (x) =  lim f (x+h) - f (x)
          h→0            h

= lim log_a (x+h) - log_ax
 h→0       h

= lim [ 1/h log_a ( x+h ) ]
h→0                    x

= lim [ 1/x . x/h log_a (1 + h/x) ]


= lim [ 1/x log_a (1 + h/x)^x/h ]
 h→0

= 1/x  lim [ log_a ( 1 +h/x)^x/h ]
        h→0

= 1/x log_a [ lim (1+ h/x)^x/h ]
               h→0

Now, write h/x = k, and note that as h→0, then  k→0 . Thus.

f ' (x) = 1/x log_a [ lim (1+k)1/k ]

= 1/x log_a e.

Hence   d/dx [ log_a x] = 1/x log_a e.

In particular, replacing a by e, we obtain

d/dx [logx] = 1/x log_a e = 1/x

Derivative of a^x, a > 0.

Let f (x) = ax, then applying the definition of the derivative, we obtain

f ' (x) = lim  f (x+h) - f (x)
         h→0     h

= lim      a^x+h - a^x
 h→0            h

= lim  a^x (a^h - 1)
 h→0     h

= lim [a^x (a^h - 1) / h]

=a^x lim [ a^h - 1 ]

= a^x log_e a

Hence    d/dx (a^x) = a^x loge a.

In particular, we have

d/dx (e^x) = e^x log_e e= e^x.

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