14 ## DERIVATIVES OF LOGARITHMIC## AND## EXPONENTIAL FUNCTIONSThe derivative of THE SYSTEM OF NATURAL LOGARITHMS has the number called e as it base; it is the system we use in all theoretical work. (In the next Lesson, we will see that e is approximately 2 We denote the logarithmic function with base e as "ln ln
In other words, this logarithm function --
-- has for its inverse the exponential function,
Here are the inverse relations: ln e And the logarithm of the base itself is always 1: ln e = 1. (Topic 20 of Precalculus.) The function
(Topic 20 of Precalculus.) Like all the rules of algebra, they will obey the rule of symmetry.
The derivative of ln We will now apply the definition of the derivative to prove:
In the course of the proof, it will make for a great simplification if we define the A limit in the proof will have that same form. Later, we will call the variable Here is the difference quotient:
We now take the limit as
We now define that limit to be the base of the natural logarithms, the number we will call e. (That limit is the one above, with Therefore,
Which is what we wanted to prove. -- that is, e, exists as
The derivative of We will now prove:
"The derivative of e is equal to e Since
Therefore on taking the derivative of both sides with respect to
e What does that imply? It implies the meaning of exponential growth. For we say that a quantity grows "exponentially" when it grows at a All exponential functions have the form
where
In the system of natural logarithms, in which
The derivative of When That is, "The derivative of is equal to
Example 1. Calculate the derivative of e
Problem 1. Calculate the derivative of e To see the answer, pass your mouse over the colored area.
e Problem 2. Calculate the derivative of the following. a) e b) e c) According to the product rule.
The derivative of ln When That is,
Example 4. Find the derivative of ln
Problem 3. Differentiate the following.
Problem 5. The derivative of log According to the rule for changing from base
Calculate the limit of that derivative a) when That derivative approaches 0, that is, becomes smaller. b) when That derivative becomes larger. The general power rule We can now prove that the derivative of
Let
That is what we wanted to prove. (If
The derivative of We will prove:
"The derivative of an exponential function with base is equal to the natural logarithm of that base times the exponential function." Let
This is what we wanted to prove.
Problem 7. Calculate the derivative of By the chain rule:
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