The exponential function is the function whose derivative is equal to its equation. In other words:
Because of this special property, the exponential function is very important in mathematics and crops up frequently.
Note: You may choose to leave this portion unheeded, for knowledge of this may not be required by your syllabus.
We shall begin with the differentiation of a simple exponential function - - from the first principles. If you're unfamiliar with differentiation with first principle, might I recommend you first direct your notice yonder: Differentiation from First Principles.
From this one can ascribe veracity to the relationship . Gleaning from it one can also deduce that the value of k depends upon the value of a and for the value of k to be equal to 1, the value of a must be smaller than 3.
To find the value of a, where k=1, we use the same relationship.
The value that is found using values of n, as it approaches infinity gives the us the following value for a:
This value of a, one of the most beautiful and prolific numbers in all of mathematics, is known as Euler's number or as you may know it e.
Putting the value of a equal to e, and thence you can derive:
This method of proving the derivative of is , though simpler, requires one to draw from the chain rule and the differentiation of . A revision and the proper derivation of the function can be found below.
The Chain Rule:
If such a case can be made the chain rule declares. (Note: these should not be thought of as fractions)
In this case:
One of the logarithmic properties dictates that the power of the number of whose logarithm is to be find, can be brought down to be multiplied with the rest of the function.
Hence it follows that the above function can be written as:
The differential can also be found using Chain Rule.
The Natural Logarithm
It therefore follows that:
This article is not detailed enough for all you need to know for C3 modules.