text: Home
  
  #### The Six Pillars of Calculus
  
  A picture is worth 1000 words
  
  #### Trigonometry Review
  
  The basic trig functions
  Basic trig identities
  The unit circle
  Addition of angles, double and half angle formulas
  The law of sines and the law of cosines
  Graphs of Trig Functions
  
  #### Exponential Functions
  
  Exponentials with positive integer exponents
  Fractional and negative powers
  The function $f(x)=a^x$ and its graph
  Exponential growth and decay
  
  #### Logarithms and Inverse functions
  
  Inverse Functions
  How to find a formula for an inverse function
  Logarithms as Inverse Exponentials
  Inverse Trig Functions
  
  #### Intro to Limits
  
  Close is good enough
  Definition
  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
  Why do these laws work?
  Two limit theorems
  How to algebraically manipulate a 0/0?
  Limits with fractions
  Limits with Absolute Values
  Limits involving Rationalization
  Limits of Piece-wise Functions
  The Squeeze Theorem
  
  #### Continuity and the Intermediate Value Theorem
  
  Definition of continuity
  Continuity and piece-wise functions
  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
  
  In Physics
  In Economics
  In Biology
  
  #### Related Rates
  
  Overview
  How to tackle the problems
  
  #### Linear Approximation and Differentials
  
  Overview
  Examples
  An example with negative $dx$
  
  #### Differentiation Review
  
  Basic 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
  
  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
  
  ### The derivative of $e^x$
  
  The function $f(x) = e^x$ is quite peculiar: it is the only function (up to scale) whose derivative is itself. We have \begin{eqnarray*} f'(x) & = & \lim_{h \to 0} \frac{f(x+h)-f(x)}{h} \cr & = & \lim_{h \to 0} \frac{e^{x+h} - e^x}{h} \cr & = & \lim_{h \to 0} \frac{e^xe^h - e^x}{h} \cr & = & e^x \cdot\left(\lim_{h \to 0} \frac{e^h - 1}{h}\right). \end{eqnarray*} The last step was made possible by the fact that $e^x$ doesn't depend on $h$. Notice that $$\lim_{h \to 0} \frac{e^h - 1}{h}=f'(0),$$ which we have defined to be 1 earlier. This implies that the derivative of $e^x$ is $e^x$!
explanation: The document is a detailed explanation of the derivative of the exponential function \( e^x \). It includes a step-by-step derivation showing that the derivative of \( e^x \) is \( e^x \) itself. The document does not contain a specific question or a question-and-answer pair.
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distilabel_metadata.content: Home
  
  #### The Six Pillars of Calculus
  
  A picture is worth 1000 words
  
  #### Trigonometry Review
  
  The basic trig functions
  Basic trig identities
  The unit circle
  Addition of angles, double and half angle formulas
  The law of sines and the law of cosines
  Graphs of Trig Functions
  
  #### Exponential Functions
  
  Exponentials with positive integer exponents
  Fractional and negative powers
  The function $f(x)=a^x$ and its graph
  Exponential growth and decay
  
  #### Logarithms and Inverse functions
  
  Inverse Functions
  How to find a formula for an inverse function
  Logarithms as Inverse Exponentials
  Inverse Trig Functions
  
  #### Intro to Limits
  
  Close is good enough
  Definition
  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
  Why do these laws work?
  Two limit theorems
  How to algebraically manipulate a 0/0?
  Limits with fractions
  Limits with Absolute Values
  Limits involving Rationalization
  Limits of Piece-wise Functions
  The Squeeze Theorem
  
  #### Continuity and the Intermediate Value Theorem
  
  Definition of continuity
  Continuity and piece-wise functions
  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
  
  In Physics
  In Economics
  In Biology
  
  #### Related Rates
  
  Overview
  How to tackle the problems
  
  #### Linear Approximation and Differentials
  
  Overview
  Examples
  An example with negative $dx$
  
  #### Differentiation Review
  
  Basic 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
  
  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
  
  ### The derivative of $e^x$
  
  The function $f(x) = e^x$ is quite peculiar: it is the only function (up to scale) whose derivative is itself. We have \begin{eqnarray*} f'(x) & = & \lim_{h \to 0} \frac{f(x+h)-f(x)}{h} \cr & = & \lim_{h \to 0} \frac{e^{x+h} - e^x}{h} \cr & = & \lim_{h \to 0} \frac{e^xe^h - e^x}{h} \cr & = & e^x \cdot\left(\lim_{h \to 0} \frac{e^h - 1}{h}\right). \end{eqnarray*} The last step was made possible by the fact that $e^x$ doesn't depend on $h$. Notice that $$\lim_{h \to 0} \frac{e^h - 1}{h}=f'(0),$$ which we have defined to be 1 earlier. This implies that the derivative of $e^x$ is $e^x$!
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document is a detailed explanation of the derivative of the exponential function \( e^x \). It includes a step-by-step derivation showing that the derivative of \( e^x \) is \( e^x \) itself. The document does not contain a specific question or a question-and-answer pair.
  
  Label: no-question