Now that you have studied the Exponential Functions and their Derivatives, you need to complete the process by looking at their inverses, the Logarithmic Functions. You will start by looking at the Natural Log because it will give you the tool to solve the other.
In the last section you used an exponentiation trick to help discover the derivative of the general exponential function. This same method is used to derive the derivative of the Natural Log Function. You know from Section 4.1 that any expression can be equated to the exponent of the Natural Log Function so let's do just that with x.
Using this identity, Example 4.6 will derive the derivative of the Natural Log Function.
Example 4.6 The Derivative of
the Natural Log Function
LiveMath will want to solve the RHS imediately, so to slow it down, you will manipulate the equation in a special way.
What happened here is that LiveMath took the derivative of both sides. Since the independent variable is x, it had no trouble solving the LHS, but since the RHS is a function of u, the chain rule was used. The equation awaits the substitution of u back into the equation. NOT yet though! Before you do that, you must manipulate the equation in the following manner.
Backwards, for sure, but the correct answer, never the less!
The general rule can be stated now for Natural Logs of Functions.
The next example will use LiveMath to demonstrate the technique for you.
Example 4.7 Natural Log Differentiation
of a Function
Of course, LiveMath can do this in one step for you.
