Proofs of derivatives of ln(x) and e^x | Taking derivatives | Differential Calculus | Khan Academy

Khan Academy
2 Oct 200912:26
EducationalLearning
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TLDRThe video script presents a detailed mathematical proof of the derivative of the natural log of x, demonstrating that it is equal to 1/x. The creator clarifies misconceptions about circularity in previous proofs by carefully avoiding the assumption of the derivative of e^x before proving it. The proof uses logarithm properties and a chain rule approach, ultimately showing that the derivative of e^x is e^x itself, without circular reasoning.

Takeaways
  • 📝 The video script is a detailed mathematical proof aimed at clarifying previous misconceptions about the derivative of natural log(x) and e^x.
  • 🔄 The author addresses accusations of circular reasoning in their initial proof and aims to provide a more rigorous and clear explanation in this video.
  • 🌟 The proof begins with the derivative of natural log(x), using the basic definition of a derivative and logarithmic properties without assuming the result beforehand.
  • 🎯 A key step in the proof involves rewriting the expression using logarithmic properties and making a substitution to simplify the equation further.
  • 🔢 The author makes a crucial substitution, replacing delta x/x with 1/n, which allows for the introduction of the number e through its definition, without circularity.
  • 🌐 The proof leverages the definition of e as the limit of (1 + 1/n)^n as n approaches infinity, which is a standard mathematical definition not reliant on the derivative of e^x.
  • 🔄 The second part of the video tackles the derivative of e^x, using both simplification and the chain rule to arrive at the result, again without assuming the derivative of e^x beforehand.
  • 📈 The chain rule application involves finding the derivative of the inner function (e^x) and then multiplying it by the derivative of the outer function (natural log(x))
  • 💡 The video concludes with the proof that the derivative of e^x is e^x, which is stated for the first time in the explanation, solidifying the non-circular nature of the proof.
  • 🌟 The author emphasizes the importance of clarity in mathematical proofs and acknowledges the responsibility of the presenter to avoid any ambiguity or misinterpretation.
  • 📚 The script serves as an educational resource for those looking to understand the derivatives of natural log(x) and e^x with rigor and precision.
Q & A

  • What was the main concern regarding the original proof of the derivative of the natural log of x?

    -The main concern was that some people felt the original proof was circular, meaning it used the result it was trying to prove.

  • How did the speaker address the concern of circularity in their new proof?

    -The speaker addressed the concern by providing a more detailed proof that clearly separates the definitions and properties used, ensuring that the result is not assumed before it is shown to be true.

  • What is the basic definition of a derivative used in the proof?

    -The basic definition of a derivative used in the proof is the limit as delta x approaches 0 of (f(x + delta x) - f(x)) / delta x.

  • How did the speaker utilize logarithm properties in the proof?

    -The speaker used logarithm properties to rewrite the expression for the derivative of the natural log of x in a form that would allow them to make a substitution to simplify the proof.

  • What substitution did the speaker make to simplify the proof?

    -The speaker made the substitution of delta x / x with 1 / n, and then took the limit as n approaches infinity, which is equivalent to delta x approaching 0.

  • What is the definition of e used in the proof?

    -The definition of e used in the proof is the limit as n approaches infinity of (1 + 1/n)^n.

  • How did the speaker prove the derivative of the natural log of x without assuming the derivative of e?

    -The speaker used the definition of e and the properties of logarithms to show that the derivative of the natural log of x is 1/x without directly assuming the derivative of e.

  • What are the two methods the speaker used to find the derivative of e to the x?

    -The speaker used both simplification by canceling out terms and the chain rule to find the derivative of e to the x.

  • What was the final result of the derivative of e to the x?

    -The final result of the derivative of e to the x is e to the x, which was derived without assuming the result beforehand.

  • How did the speaker conclude that the proof is not circular?

    -The speaker concluded that the proof is not circular because they derived the result without assuming the derivative of e to the x at any point during the proof.

  • What was the speaker's goal in providing this new proof?

    -The speaker's goal was to clarify the original proof and provide a more detailed explanation to address the concerns of circularity and to ensure a clear understanding of the concepts involved.

Outlines
00:00
📚 Proof of Derivative of Natural Logarithm

The paragraph begins with the speaker addressing a previous video where they proved the derivative of the natural logarithm of x. They clarify that their initial proof was not circular and aim to redo the proof with more sophisticated tools. The speaker starts by defining the derivative of the natural log of x using the basic definition of a derivative. They use logarithm properties to simplify the expression and make a substitution to eventually relate it to the number e, without assuming the derivative of e beforehand. The goal is to show that the derivative of the natural log of x is 1/x, without circularity in the argument.

05:03
📈 Simplified Derivative of Natural Logarithm

In this paragraph, the speaker continues the process of simplifying the derivative of the natural logarithm. They use exponent properties to rearrange the expression and isolate terms. The speaker then makes a substitution to relate the expression to the definition of e, the limit of (1 + 1/n)^n as n approaches infinity. By using the definition of e and the natural logarithm (log base e), the speaker concludes that the derivative of the natural log of x equals 1/x. They emphasize that this result is derived strictly from the definitions, without assuming the derivative of e or the natural log of e to the x.

10:04
🔄 Derivative of Exponential Function

The speaker moves on to the derivative of the exponential function e to the x. They present two methods for finding the derivative: one by direct simplification using the natural log and the other by applying the chain rule. In both cases, the speaker arrives at the conclusion that the derivative of e to the x is equal to e to the x. They clarify that this result is not assumed but derived from the properties of the natural logarithm and the chain rule. The speaker emphasizes that the proof is not circular and that the derivative of e to the x is established independently of any prior assumptions about the derivative.

Mindmap
Keywords
💡Derivative
In the context of the video, 'derivative' refers to a fundamental concept in calculus that represents the rate of change of a function at a particular point. It is used to analyze the behavior of functions, such as their slope or rate of growth. The video script focuses on proving the derivatives of specific functions, namely the natural log of x and e to the power of x, using limits and logarithmic properties without assuming the results beforehand.
💡Natural Logarithm
The natural logarithm, often denoted as 'ln', is a logarithm with the mathematical constant 'e' as its base. 'e' is a significant number in mathematics, approximately equal to 2.71828, and the natural logarithm is a key concept in the video. The video script involves a detailed proof of the derivative of the natural log of x, which is shown to be 1/x, using limit concepts and properties of logarithms.
💡Limit
A 'limit' in mathematics is a value that a function or sequence 'approaches' as the input (or index) gets arbitrarily close to a certain point. In calculus, limits are used to define derivatives and integrals. The video script utilizes limits to define derivatives and prove theorems without making circular assumptions.
💡Logarithm Properties
Logarithm properties are rules that govern the behavior of logarithms and allow for the simplification and manipulation of logarithmic expressions. These properties include the product rule (log(a) + log(b) = log(a*b)), the quotient rule (log(a) - log(b) = log(a/b)), and the power rule (n * log(a) = log(a^n)). In the video, these properties are used to rewrite and simplify expressions while proving derivatives.
💡Exponential Function
An exponential function is a mathematical function of the form f(x) = a^x, where 'a' is a constant and 'x' is the variable. Exponential functions are important in many areas of mathematics and science as they model growth processes. In the video, the focus is on the derivative of an exponential function with base 'e', which is a central concept in calculus.
💡Chain Rule
The chain rule is a fundamental rule in calculus used to find the derivative of a composite function. It states that the derivative of a function composed of two or more functions is the derivative of the outer function times the derivative of the inner function. The video uses the chain rule to prove the derivative of the natural log of e to the x, without assuming the derivative of e to the x.
💡Circular Proof
A circular proof, or circular reasoning, occurs when a conclusion is based on premises that the conclusion itself is supposed to prove. This is a logical fallacy because it doesn't actually provide evidence for the conclusion. The video script addresses this by carefully structuring the proof to avoid assuming the very thing being proven.
💡e (Euler's number)
Euler's number, commonly denoted as 'e', is a significant mathematical constant approximately equal to 2.718281828459. It is the base of the natural logarithm and has extensive applications in calculus, particularly in the study of exponential growth and continuous compound interest. The video script discusses 'e' in the context of proving the derivative of e to the x without circular reasoning.
💡Proof
In mathematics, a 'proof' is a logical argument that demonstrates the truth of a statement or theorem. Proofs are essential for establishing the validity of mathematical concepts and theorems. The video script is focused on providing clear, non-circular proofs for the derivatives of specific functions, namely the natural log of x and e to the power of x.
💡Transcendental Functions
Transcendental functions are functions that are not algebraic, meaning they cannot be expressed using only the basic arithmetic operations and root operations. Examples include logarithmic, exponential, and trigonometric functions. In the video, both the natural log and e to the x are transcendental functions, and the derivatives of these functions are being explored.
💡Slope
The 'slope' of a line or curve at a particular point is a measure of how steep the line or curve is at that point. In calculus, the concept of slope is generalized to all functions, and it is what derivatives represent. The video script discusses the derivatives of functions, which can be thought of as the slopes of the graphs of those functions.
Highlights

The video aims to disprove the claim of circularity in a previous proof of the derivative of the natural log of x.

The proof begins with the basic definition of a derivative, using the limit as delta x approaches 0.

Logarithm properties are used to simplify the expression and rewrite it in terms of a new variable, 1/delta x.

A substitution is made, defining delta x/x as 1/n, to simplify the expression further.

The limit as n approaches infinity is taken, which is equivalent to the original limit as delta x approaches 0.

The expression is rewritten using the definition of e, without assuming the derivative of e to the x.

The derivative of the natural log of x is shown to be 1/x, using only the definition of e and logarithm properties.

The video then moves on to prove the derivative of e to the x, using two different methods.

The first method simplifies the expression, showing that the derivative is the same as the derivative of x.

The second method applies the chain rule, treating e to the x as an inner function.

The chain rule leads to an expression that must be equal to the derivative found using the first method.

By solving the resulting equation, the derivative of e to the x is found to be e to the x.

The video emphasizes that the derivative of e to the x was not assumed in the previous steps.

The proof demonstrates that the derivative of the natural log of x and e to the x can be found without circular reasoning.

The video aims to provide clarity on the issue of circularity in mathematical proofs.

The presenter acknowledges the importance of clear communication in mathematical explanations.

The video serves as an example of rigorous mathematical proof that avoids circular logic.

Transcripts

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