The exponential function is used to representation of the phenomenon when a constant change in the independent variable gives the same proportional transform (increase or decrease) in the dependent variable. The exponential function is frequently written as exp(x), especially when the input is an appearance too complex to be written as an exponent.
Formula for the Exponential E^x:
The exponential function e^x has the approximate value as 2. 718281828.
The exponential identity can be represented as,
exp( x+ y) = exp ( x) + exp (y).
The differentiation of the exponential function e^x is the same that means e^x.
The value for e^0 is 1.
The function e^ (a + ib) can be expressed as e^a ( cos b + i sin b)
The differentiation for the e^ f(x) is obtained by using the chain rule that can be represented as
D( e^ f(x) )= f(x) e^ f(x).
The power series of the exponential is expressed as
e^x = 1+ x+ (x^2 /2 ! ) + ( x^3 / 3! ) +
Examples for the Exponential E^x:
Example 1:
Find the differentiation for e^2x.
Solution:
The given function is e^2x.
The given function is differentiated with respect to x,
Then the function is represented as 2 e^ 2x.
The value of the differentiation for the function e^2x is 2 e^ 2x.
Example 2:
Find the value for e^3x using the exponential power series.
Solution:
The given function is e^3x.
The formula for the power series of the exponential form is