## Derivatives of logarithmic And Exponential

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# Derivatives of logarithmic And Exponential

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

Let

f (x) = log

Thens f ' (x) =

h→0 h

= lim

h→0 h

= lim [ 1/h log

h→0 x

= lim [ 1/x . x/h log

= lim [ 1/x log

h→0

= 1/x lim [ log

h→0

= 1/x log

h→0

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

f ' (x) = 1/x log

= 1/x log

Hence d/dx [ log

In particular, replacing a by e, we obtain

d/dx [logx] = 1/x log

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

f ' (x) = lim

h→0 h

= lim

h→0 h

= lim

h→0 h

= lim [a

=a

= a

Hence

In particular, we have

d/dx (e

For more help in

Derivative of log

_{a}x.Let

f (x) = log

_{a}xThens f ' (x) =

__lim f (x+h) - f (x)__h→0 h

= lim

__log___{a}(x+h) - log_{a}xh→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}aHence

**d/dx (a**^{x}) = a^{x}loge a.In particular, we have

d/dx (e

^{x}) = e^{x}log_{e}e= e^{x}.For more help in

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