5. Differentiating exponentials and logarithms
Differentiating exponential functions
An exponential function is simply a function with an exponent in it; in other terms, a function of the form: 
.
Numerical estimates can be used to approximate the gradient at a point, and this can be generalised:
Substituting the function in:
.
So the gradient is merely a multiple of the original function.
The number e
The number "e" arises a lot in mathematics, and can be calculated:
.
It has a special property in the context of this differential method, and to discover this it is useful to consider the factor of multiplication of "e" that there would be (using the previously defined formula) for the derivative of 
. Calculating this will give something that tends to 1, and therefore:
This function is the only function whose derivative is itself.
Another definition of this function is used:
.
Index laws can often be used to differentiate the slightly more complicated functions, for example:
.
(Alternatively, the method discussed in the previous section of the notes can be used).
As differentiation and integration are opposites, it is simple to see the integral of the exponential function:
.
Application to exponential growth and decay
The natural logarithm is often used in mathematics, this is the logarithm with base "e"; it is used due to the special properties that it has.
.
As it is usual to differentiate using "e", it might be necessary to write the equation which models the exponential growth using this constant as a base. This is easier to do (often) using the logarithmic form:
So, in this case the gradient of the graph of 
against "t" is "c".
Differentiating the natural logarithm
The exponential function (with base "e") is the inverse function of the natural logarithm, and as they are both one-one, they are reflections of each other in the line 
with their natural domain.
Consider a point (a, b) on the curve 
; this can be represented also as 
.
Due to the reflection, the corresponding point on the natural logarithm curve is: 
.
Consider any line 
, and a corresponding line which is the reflection in the line .
Hence: 
.
Hence the new gradient is 
.
Hence, the gradient of the given point on the logarithmic curve is 
. This is true for all values of "a" on the curve, and therefore:
.
The reciprocal integral
Integrating the powers of "x" is a basic task, however there is a problem with a certain power:
.
Using the rule of:
Produces a problem, as there is division by zero for this value. Again, results from differentiation are helpful; due to the relationship between differentiation and integration, it is simple to say:
.
Extending the reciprocal integral
The natural logarithm is not defined for values 
, however, 
is defined.
Testing the derivative of the natural logarithm of a negative "x" can see if this method of making all negative values positive (by changing signs):
.
Hence the modulus function can be used:
.
The derivative of 
Before it was seen that the derivative of the general exponential function was 
where the constant "c" was to be determined.
Consider:
Hence:
.
Irrational indices
Core 2 does not consider the irrational indices, however they can be considered.
Using the reciprocal integral as a definition of the natural logarithm, only an irrational limit of integration is needed which is acceptable. Due to the fact that the product of two reals is real, and also that the exponential function is defined over the reals, there is always a real value for an irrational index.
