revision notes .pdf : exponentials,logs
Exponential functions
Strictly speaking all functions where the variable is in the index are called exponentials.
The Exponential function ex
This is the one particular exponential function where 'e' is approximately 2.71828 and the gradient of y= ex at (0,1) is 1.
One other special quality of y= ex is that its derivative is also equal to ex
and for problems of the type y= ekx
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=logex means:
x is the number produced when e is raised to the power of y
The connection between y=ex and y=logex can be shown by rearranging y=logex.
y=logex can be written as x=ey
(logex 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(x3+3)
Problems of the type y=Nf(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=10x
Example #2
find the derivative of y= ln(cos32x)
A graphical comparison of exponential and log functions
As you can see, y= ex is reflected in the line y=x to produce the curve y=ln(x)
