Exponential functions
Strictly speaking all functions where the variable is in the index are called exponentials.
The Exponential function e^{x}
This is the one particular exponential function where 'e' is approximately 2.71828 and the gradient of y= e^{x} at (0,1) is 1.
One other special quality of y= e^{x} is that its derivative is also equal to e^{x}
and for problems of the type y= e^{kx}
Derivative problems like the above concerning 'e' are commonly solved using the Chain Rule.
Example #1
Find the derivative of:
Example #2
find the derivative of:
Derivative of a Natural Logarithm function
Remember y=log_{e}x means:
x is the number produced when e is raised to the power of y
The connection between y=e^{x} and y=log_{e}x can be shown by rearranging y=log_{e}x.
y=log_{e}x can be written as x=e^{y}
(log_{e}x is now more commonly written as ln(x) )
The derivative of ln(x) is given by:
Example #1
find the derivative of y = ln(3x)
Example #2
find the derivative of y = ln(x^{3}+3)
Problems of the type y=N^{f(x)}
Problems of this type are solved by taking logs on both sides and/or using the Chain Rule.
Example #1
find the derivative of y=10^{x}
Example #2
find the derivative of y= ln(cos^{3}2x)
A graphical comparison of exponential and log functions
As you can see, y= e^{x} is reflected in the line y=x to produce the curve y=ln(x)

