Limit Laws and Computations

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$e^x$

Product and Quotient Rules

The Product Rule
The Quotient Rule

Derivatives of Trig Functions

Necessary Limits
Derivatives of Sine and Cosine
Derivatives of 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
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

How to take derivatives
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
Critical Numbers
Steps to Find Absolute Extrema

The Mean Value and other Theorems

Rolle's Theorems
The Mean Value Theorem
Finding

$c$

$f$ vs. $f'$

Increasing/Decreasing Test and Critical Numbers
Process for finding intervals of increase/decrease
The First Derivative Test
Concavity
Concavity, Points of Inflection, and the Second Derivative Test
The Second Derivative Test
Visual Wrap-up

Indeterminate Forms and L'Hospital's Rule

What does

$\frac{0}{0}$ equal?
Examples
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

Antiderivatives
Common antiderivatives
Initial value problems
Antiderivatives are not Integrals

The Area under a curve

The Area Problem and Examples
Riemann Sum Notation
Summary

Definite Integrals

Definition of the Integral
Properties of Definite Integrals
What is integration good for?
More Applications of Integrals

The Fundamental Theorem of Calculus

Three Different Concepts
The Fundamental Theorem of Calculus (Part 2)
The Fundamental Theorem of Calculus (Part 1)
More FTC 1

How to algebraically manipulate a limit of the form $\tfrac00$

Factoring can help us evaluate a limit. For example, we already looked at the limit

$\lim_{x \to 1} \frac{x^2-1}{x-1}$ Since

$x^2-1 = (x-1)(x+1)$ , we have

$\lim_{x \to 1} \frac{x^2-1}{x-1} = \lim_{x \to 1} \frac{(x-1)(x+1)}{x-1} = \lim_{x\to 1} (x+1) = 2.$ Canceling the factors of

$x-1$ from the numerator and denominator is OK for all values of

$x$ except $x=1$ . However, we're only looking at what happens when

$x$ is close to 1, not equal to 1, so the cancellation makes sense.

The Six Pillars of Calculus

Trigonometry Review

Exponential Functions

Logarithms and Inverse functions

Intro to Limits