The Derivative Function

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Trigonometry Review

The basic trig functions
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Logarithms as Inverse Exponentials
Inverse Trig Functions

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Infinite Limits
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Limit Laws
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Definition of continuity
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Vertical asymptotes (Redux)
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Rates of Change

Average velocity
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Computing an instantaneous rate of change of any function
The equation of a tangent line
The Derivative of a Function at a Point

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
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Derivatives of Trig Functions

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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
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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

How to take derivatives
Basic Building Blocks
Advanced Building Blocks
Product and Quotient Rules
The Chain Rule
Combining Rules
Implicit Differentiation
Logarithmic Differentiation
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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

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$f$ vs. $f'$

Increasing/Decreasing Test and Critical Numbers
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The First Derivative Test
Concavity
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The Second Derivative Test
Visual Wrap-up

Indeterminate Forms and L'Hospital's Rule

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

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Newton's Method

The Idea of Newton's Method
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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

The Derivative Function

We have already seen that the derivative of a function $f$ at $x=a$ is

$f'(a) = \lim_{x \to a}\frac{f(x)-f(a)}{x-a} = \lim_{h \to 0} \frac{f(a+h)-f(a)}{h}.$
We can apply the same idea to get a formula for the derivative of

$f$ at every point:

$f'(\Box) = \lim_{h \to 0} \frac{f(\Box+h)-f(\Box)}{h},$ where

$\Box$ stands for "anything you like". Recall that we said you will need to be able to compute this second form of the derivative, with the limit as

$h\to 0$ . That is the only form we can use to compute the derivative function.

In particular, if we take $x$ as input and associate to it $f'(x)$ as output, then we have a function. We see how to compute $f'(x)$ in the video. Fortunately, you already know how to do this, as you will see.