The Chain Rule

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The Six Pillars of Calculus

The Pillars: A Road Map
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Trigonometry Review

The basic trig functions
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Exponential Functions

Exponentials with positive integer exponents
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$f(x)=a^x$ and its graph
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Logarithms and Inverse functions

Inverse Functions
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Logarithms as Inverse Exponentials
Inverse Trig Functions

Intro to Limits

Close is good enough
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One-sided Limits
How can a limit fail to exist?
Infinite Limits and Vertical Asymptotes
Summary

Limit Laws and Computations

A summary of Limit Laws
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Two limit theorems
How to algebraically manipulate a 0/0?
Limits with fractions
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Limits of Piece-wise Functions
The Squeeze Theorem

Continuity and the Intermediate Value Theorem

Definition of continuity
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Continuity properties
Types of discontinuities
The Intermediate Value Theorem
Examples of continuous functions

Limits at Infinity

Limits at infinity and horizontal asymptotes
Limits at infinity of rational functions
Which functions grow the fastest?
Vertical asymptotes (Redux)
Toolbox of graphs

Rates of Change

Tracking change
Average and instantaneous velocity
Instantaneous rate of change of any function
Finding tangent line equations
Definition of derivative

The Derivative Function

The derivative function
Sketching the graph of

$f'$
Differentiability
Notation and higher-order derivatives

Basic Differentiation Rules

The Power Rule and other basic rules
The derivative of

$e^x$

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

Related Rates

Overview
How to tackle the problems
Example (ladder)
Example (shadow)

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
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The First Derivative Test
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Indeterminate Forms and L'Hospital's Rule

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

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Another Example

Newton's Method

The Idea of Newton's Method
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When NM doesn't work

Anti-derivatives

Anti-derivatives and Physics
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Anti-derivatives are not Integrals

The Area under a curve

The Area Problem and Examples
Riemann Sums Notation
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Definite Integrals

Definition
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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

A hybrid Chain Rule

In addition to Versions 1 and 2 of the chain rule, there is also a useful hybrid form:

$\frac{d}{dx}f(u) = f'(u) \frac{du}{dx}$

Applications

This form allows us to expand the scope of many of our derivative formulas:

The derivative of $x^n$ (with respect to $x$ ) is $n x^{n-1}$ , so the derivative of $u^n$ is $n u^{n-1} \frac{du}{dx}$ .
The derivative of $\sin(x)$ is $\cos(x)$ , so the derivative of $\sin(u)$ is $\cos(u) \frac{du}{dx}$ .
The derivative of $\cos(x)$ is $-\sin(x)$ , so the derivative of $\cos(u)$ is $-\sin(u) \frac{du}{dx}$ .
The derivative of $e^x$ is $e^x$ , so the derivative of $e^u$ is $e^u \frac{du}{dx}$ .

These formulas come up a lot, and are worth memorizing.

We can also use the chain rule to go back and find derivatives of some functions we couldn't derive before.

Example: What is the derivative of

$a^x$ , where

$a$ is a positive number?

Solution: Since

$a= e^{\ln(a)}$ , we can rewrite

$a^x$ as

$(e^{\ln(a)})^x = e^{x \ln(a)}$ . Taking

$u=x \ln(a)$ , we have

$\begin{eqnarray*} \displaystyle{\frac{d a^x}{dx}} &=& \displaystyle{\frac{d}{dx}(e^{x \ln(a)})} \cr &=& e^{x \ln(a)} \displaystyle{\frac{d}{dx}}\Big(x\ln(a)\Big)\cr & = & e^{x \ln(a)} \ln(a) \cr & = & a^x \ln(a). \end{eqnarray*}$ (Notice that

$\ln(a)$ is a constant.)