Logs and Derivatives

Definition of the Natural Logarithm

Recall that

What is

Definition:  For x > 0 we define

Note:
The Second Fundamental Theorem of Calculus tells us that

d/dx (ln x) = 1/x

Properties of ln x

1. ln 1 = 0

2. ln(ab) = ln a + ln b

3. ln(an) = n ln a

4. ln(a/b) = ln a - ln b

Proof of (3)

So that

ln(xn

and

n ln x

have the same derivative.  Hence

ln(xn) = n ln x + C

Plugging in x = 1 we have that C = 0.

Definition of e

Let e be such that

ln e = 1

ie.

Examples and Exercises

Example

Find the derivative of

ln (x2  + 1)

Solution

We use the chain rule with

y = ln u,    u = x2 + 1

2x
y'  =  (2x)(1/u)  =
x2 + 1

Find the derivatives of the following functions:

1. ln (lnx)

2. (ln x)/x

3. (ln x)2

4. ln (sec x)

5. ln (csc x)

6. Show that

3 ln x - 4

is a solution of the differential equation

xy'' + y' = 0

7. Find the relative extrema of

x ln x

8. Find the equation of the tangent line to

y = 3x2 - ln x

at (1,3)

9. Find dy/dx for

ln(xy) + 2x2 = 30