Section 5.1 The Natural Log

Function:Differentiation

Section 5.1 The Natural Log Function: Differentiation

Since the winter break we have spent our mental energy on integrating functions. We have looked at the integral as an infinite sum, as an antiderivative of a function, and as an area bounded by a curve, the x-axis, and vertical boundaries.

Now, we will step back and look at differentiation rules for logarithmic and exponential functions before we deal with antidifferentiating them.

Section 5.1 The Natural Log Function: Differentiation

You are familiar with the power rule for integrals which states that

This definition will clearly cause us problems if we try to apply it to the simple function of

1

1

nn xx dx C

n

1y

x

Section 5.1 The Natural Log Function: Differentiation

We need a new definition to deal with this function.

Let’s take out our calculator and see if we can verify this definition. Any ideas on how to do so?

1

1ln

x

x dtt

Section 5.1 The Natural Log Function: Differentiation

We should remind ourselves of the important properties of the natural logarithm function as well as remember where we have used the function in the past. The Laws of Logarithms:

ln ln ln( )

ln ln ln

ln lnb

a b ab

aa b

b

a b a

Section 5.1 The Natural Log Function: Differentiation

We should also be reminded of a special number that comes into play when dealing with natural logarithms. That number is called e. You have seen a definition of e in the past. That definition is

We have a new definition now

1

11

e

dxx

11lim

n

n

en

Section 5.1 The Natural Log Function: Differentiation

We can use the properties of logarithms to help clean up some ugly looking differentiation problems in addition to practicing new integrals based on our definitions. The Fundamental Theorem of Calculus tells us that

1

1 1ln ln

x ddt x xt dx x

Section 5.1 The Natural Log Function: Differentiation

Adding the Chain Rule to that application of the Fundamental Theorem of Calculus we have the following derivative statement and an example of its application:

1ln

d duu

dx u dx

22

2 3ln 3 1

3 1

dy xy x x

dx x x

Section 5.1 The Natural Log Function: Differentiation

Here are a couple of more examples to try. See how you might use the properties of logarithms to help simplify the differentiation process.

31

1

x dyy

x dx

2

2

3 2

1

x x dyy

dxx

3 lndy

y x xdx