Inverse Functions and Logarithms

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$e^x$

Product and Quotient Rules

The Product Rule
The Quotient Rule

Derivatives of Trig Functions

Two important Limits
Sine and Cosine
Tangent, Cotangent, Secant, and Cosecant
Summary

The Chain Rule

Two forms of the chain rule
Version 1
Version 2
Why does it work?
A hybrid chain rule

Implicit Differentiation

Introduction and Examples
Derivatives of Inverse Trigs via Implicit Differentiation
A Summary

Derivatives of Logs

Formulas and Examples
Logarithmic Differentiation

Derivatives in Science

In Physics
In Economics
In Biology

Linear Approximation and Differentials

Overview
Examples
An example with negative

$dx$

Differentiation Review

Basic Building Blocks
Advanced Building Blocks
Product and Quotient Rules
The Chain Rule
Combining Rules
Implicit Differentiation
Logarithmic Differentiation
Conclusions and Tidbits

Absolute and Local Extrema

Definitions
The Extreme Value Theorem
Fermat's Theorem
How-to

The Mean Value and other Theorems

Rolle's Theorems
The Mean Value Theorem
Finding

$c$

$f$ vs. $f'$

Increasing/Decreasing Test and Critical Numbers
How-to
The First Derivative Test
Concavity, Points of Inflection, and the Second Derivative Test

Indeterminate Forms and L'Hospital's Rule

What does

$\frac{0}{0}$ equal?
Indeterminate Differences
Indeterminate Powers
Three Versions of L'Hospital's Rule
Proofs

Optimization

Strategies
Another Example

Newton's Method

The Idea of Newton's Method
An Example
Solving Transcendental Equations
When NM doesn't work

Anti-derivatives

Anti-derivatives and Physics
Some formulas
Anti-derivatives are not Integrals

The Area under a curve

The Area Problem and Examples
Riemann Sums Notation
Summary

Definite Integrals

Definition
Properties
What is integration good for?
More Examples

The Fundamental Theorem of Calculus

Three Different Quantities
The Whole as Sum of Partial Changes
The Indefinite Integral as Antiderivative
The FTC and the Chain Rule

Logarithms as Inverse Exponentials

Throughout suppose that $a>1$ . The function $y=\log_a(x)$ is the inverse of the function $y=a^x$ . In other words,

$\log_a(a^x) = x \qquad \hbox{and} \qquad a^{\log_a(x)}=x$

whenever these make sense.

Examples:

Since $10^3=1000$ , $\log_{10}(1000)=3$ .

Since $2^{-3}=1/8$ , $\log_2(1/8)=-3$ .

$10^{\log_{10}(17)}=17$ and $2^{\log_2(.038)}=.038$

Since $a^0=1$ , no matter what $a$ is, $\log_a(1)=0$ . In general $\log_a(x)$ will be positive for $x>1$ and negative for $x<1$ .

The domain of $a^x$ is $(-\infty, \infty)$ , so the range of $\log_a(x)$ is $(-\infty,\infty)$ .

The range of $a^x$ is $(0,\infty)$ , so the domain of $\log_a(x)$ is $(0,\infty)$ . We cannot take the log of zero or the log of a negative number.

Laws of Logarithms

The laws of logs follow from the law of exponents. Provided that $x>0$ and $y>0$ we have:

$\log_{a}(xy) = \log_a(x) + \log_a(y)$

$\log_a(x/y) = \log_a(x) - \log_a(y)$

$\log_a(x^r) = r \log_a(x)$

$\log_b(x) = \log_a(x) \log_b(a)$

The last property (also known as the change of basis formula) shows in particular that all log functions are the same, up to scale.

Graphs

Below are the graphs when $a>1$ or $0 < a<1$ .

The Six Pillars of Calculus

Trigonometry Review

Exponential Functions

Logarithms and Inverse functions

Intro to Limits

Limit Laws and Computations

Continuity and the Intermediate Value Theorem

Limits at Infinity

Rates of Change

The Derivative Function

Basic Differentiation Rules