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Exponential Function

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Derivative of Exponential Function

What is the derivative of the function _ ~a ^~x , _ where ~a is a real number?

Putting _ f( ~x )_ = _ ~a ^~x , _ f_~x* ( ~w ) _ = _ ( ~a ^~w - ~a ^~x ) ./ ( ~w - ~x )
_ = _ ~a ^~x ( ~a ^{~w - ~x} - 1 ) ./ ( ~w - ~x )
We have _ f_0* ( ~z ) _ = _ ( ~a ^~z - 1 ) ./ ~z , _ so _ f_~x* ( ~w ) _ = _ ~a ^~x f_0* ( ~w - ~x).
Now _ ~w -> ~x _ => _ ~w - ~x -> 0 , _ so _ f '( ~x ) _ = _ ~a ^~x f '( 0 )
So if we can find the derivative at _ ~x = 0 _ we will know the value of the derivative at any _ ~x .
Put _ M_~a _ #:= _ D( ~a ^~x ) ( 0 ) , _ then _ D( ~a ^~x ) ( ~x ) _ = _ ~a ^~x M_~a
 

The Number e

  Consider the graph of _ ~y _ = _ 2 ^~x _ to the left.
We can see that this is a concave function, and it is intuitively obvious that the gradient of the tangent at 0 , _ i.e. _ M_2 , _ is less than the gradient of the chord from ( 0 , 1 ) to ( 1 , 2 ). _ But the gradient of this chord is 1 , _ so _ M_2 _ < _ 1 .

Now in general the gradient of the tangent of the curve _ ~y _ = ~a ^~x _ at 0 is greater than the gradient of the chord from any point of the curve corresponding to negative ~x , to the point corresponding to zero, i.e. ( 0 , 1 ) in all cases, as ~a ^0 = 1 _ &forall. ~a.

Let _ ~n _ be any positive integer, and consider the choord from ( - 1/~n , ~a ^{-1/~n} ) to ( 0 , 1 ). _ This has gradient _ ( 1 - ~a ^{-1/~n} ) ./ ( 1/~n ) _ _ (see diagram right).
Will this ever be greater than 1? _ If so then _ 1 - 1/~n _ > _ ~a ^{-1/~n} _ _ => _ _ ( 1 - 1/~n ) ^~n _ > _ 1/~a .

In particular , for _ ~a = 3 , _ and choosing _ ~n = 6 , _ then _ ( 1 - 1/~n ) ^~n _ = _ (5/6) ^6 _ = _ 15625 ./ 46656 _ > _ 1 ./ 3 , _ so _ M_3 _ > _ 1 .

 

So we have established that _ M_2 < 1 , _ and _ M_3 > 1 . _ It is intuitively obvious (but won't be proved) that _ M_~a _ is a continuous, and increasing function of ~a , _ so there must be a real number , _ ~e _ say, between 2 and 3 for which _ M_~e _ = _ 1 , _ so

fract{d ~e ^~x,d~x} _ _ = _ _ ~e ^~x

In fact, from the above argument it can be seen that _ ( 1 - 1/~n ) ^~n _ -> _ 1 ./ ~e , _ as _ ~n -> &infty. , _ or _ ~e _ = _ lim_{~n -> &infty.} ( 1 - 1/~n ) ^{-~n} .
This could be used as a definition for ~e, but it is more usual to use the definition _ ~e _ = _ lim_{~n -> &infty.} ( 1 + 1/~n ) ^{~n} , _ which can be arrived at by considering the chords on the positive side of the graph.

Considering both types of chords it can be seen that _ _ _ _ ( 1 - 1/~n ) ^{-~n} _ < _ ~e _ < _ ( 1 + 1/~n ) ^{~n} _ _ _ _ for any positive number _ ~n.
This can be used to evaluate ~e to the required degree of accuracy.

Logarithms

The #~{logarithmic function} is the inverse of the exponential function. So the logarithmic function to the #~{base} ~a is defined by

~y _ = _ log_~a ~x _ _ <=> _ _ ~x _ = _ ~a ^~y


Note that the domain of the logarithmic function is &reals.^+.

The #~{natural logarithms} are logarithms to the base ~e and are denoted by ln:

ln ~x _ == _ log_~e ~x

We have _ ~y = ln ~x _ <=> _ ~x = ~e ^~y , _ so _ d~x/d~y = ~e ^~y = ~x , _ so _ d~y/d~x = 1/~x .

fract{d ln ~x,d~x} _ = _ fract{1,~x}

Exponential

~y = ~a ^~x _ => _ ln ~y = ~x ln ~a _ => _ 1 ./ ~y = d~x/d~y ln ~a _ => _ d~y/d~x = ~y ln ~a = ~a ^~x ln ~a

fract{d ~a ^~x,d~x} _ = _ ~a ^~x ln ~a