
# 4.1: Logs and Derivatives


## Definition of the Natural Logarithm

Recall that

$\int x^n\, dx = \dfrac{x^{n+1}}{n+1} +C.$

What is

$\int x^{-1} dx \text{?}$

Definition

For $$x > 0$$ we define

$\int_1^x \dfrac{1}{t} dt = \ln x.$

Note

The Second Fundamental Theorem of Calculus tells us that

$\dfrac{d}{dx} \ln x = \dfrac{1}{x}.$

## Properties of $$\ln x$$:

1. $$\ln 1 = 0$$
2. $$\ln (ab) = \ln a + \ln b$$
3. $$\ln(a^n) = n \ln a$$
4. $$\ln \left( \dfrac{a}{b} \right) = \ln a - \ln b$$

Proof of (3)

$\dfrac{d}{dx} \ln x^n = \dfrac{d}{dx} \int_1^{x^n} \dfrac{1}{t} dt$

$= \dfrac{1}{x^n} \left( n\,x^{n-1}\right) = n \left(\dfrac{1}{x} \right)$

$= n \dfrac{d}{dx} \int _1^x \dfrac{1}{t} dt = n\ln x.$

So that

$\ln x^n$

and

$n \ln x$

have the same derivative. Hence

$\ln x^n = n \ln x + C.$

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

## Definition of $$e$$

Let $$e$$ be such that

$\ln e = 1$

ie.

$\int_1^e \dfrac{1}{t} dt = 1.$

Example 1

Find the derivative of

$\ln (x^2 + 1).$

Solution

We use the chain rule with $$y = \ln u$$ so $$u = x^2 + 1$$,

$y' = (2x)(1/u) = \dfrac{2x}{x^2+1}.$

Exercise

Find the derivatives of the following functions:

• $$\ln (\ln x)$$

• $$\dfrac{\ln x}{x}$$

• $$(\ln x)^2$$

• $$\ln (\sec x)$$

• $$\ln (\csc x)$$

Exercise

1. Show that $$y= 3 \ln x - 4$$ is a solution of the differential equation $$xy'' + y' = 0$$.
2. Find the relative extrema of $$x \ln x$$.

3. Find the equation of the tangent line to $$y = 3x^2 - \ln x$$ at $$(1,3)$$.

4. Find $$\dfrac{dy}{dx}$$ for $$\ln (xy) + 2x^2 = 30$$.

Larry Green (Lake Tahoe Community College)

• Integrated by Justin Marshall.