The exponential function f(x) = e^{x} is a special function. It is the only function which is its own derivative.
Some people write e^{x} as exp(x).
e is an irrational number approximately equal to 2.718282 (to 6 d.p.)
This means that at any point on the graph of the exponential function the gradient of the curve at that point is equal to the value of the function (the y-value) at that point.
The function can also be written as a series.
It can be seen that if this series is differentiated term by term the result is the same series!
If x = 1 then the value of e can be worked out to the required accuracy using the series.
This means that:
If f(x) = e^{x} then f '(x) = e^{x} |
Using the Chain Rule with Exponential Functions
To differentiate functions such as f(x) = e^{2x} the chain rule is used.
example f(x) = e^{2x}
f '(x) = 2. e^{2x} = 2e^{2x}
In general terms:
If f(x) = e^{ax} then f '(x) = ae^{ax} |
example f(x) = 3e^{200x}
f '(x) =3e^{200x} . 200
= 600e^{200x}