Natural Logs and Differentiation

Chad Gilliland
18 Nov 201310:59
EducationalLearning
32 Likes 10 Comments

TLDRThis educational video explores the concept of natural logarithms, emphasizing their definition as the area under the curve of 1/T from 1 to X, with base E. It explains the properties of natural logs, such as their graph characteristics, domain, range, and how they're always increasing and concave down. The instructor demonstrates logarithm properties, simplification techniques, and how to find derivatives using the fundamental theorem of calculus, showcasing examples and the chain rule. The video concludes with solving a differential equation using natural log derivatives, offering a comprehensive guide for students.

Takeaways
  • 📚 The natural log, also known as the logarithm with base e, is defined as the area under the curve of 1/T from 1 to T and is only defined for positive values of X.
  • 🔢 The base of the natural log is the number e, named after Leonard Euler, and e and natural log are inverse functions of each other.
  • 📈 The graph of the natural log of X has an x-intercept at (1, 0), is always increasing, and is always concave down with a domain of all positive numbers and a range of all real numbers.
  • 🌐 The natural log of 1 is 0, and the natural log of e is 1, with several other properties of logarithms that are important to remember, such as how to handle multiplication, division, and exponentiation in logarithmic expressions.
  • 🔑 The fundamental theorem of calculus helps in finding the derivative of the natural log, which is 1/X, indicating that the slope of the natural log graph is the reciprocal of the x-coordinate.
  • 🔄 The derivative of a natural log function can be simplified using the chain rule, where the derivative of the inside function is divided by the inside function itself (u'/u).
  • ✅ The product rule is used when differentiating expressions involving the natural log of a product, such as x times the natural log of x, resulting in 1 + the natural log of X.
  • 🔄 The chain rule is essential for differentiating more complex expressions involving natural logs, such as the natural log of the square root of x plus 1, which simplifies to 1/2 times the derivative of (x + 1)^(1/2).
  • 🔍 The natural log of the natural log of X can be differentiated using u-substitution, where u equals the natural log of X, resulting in a derivative of 1/X times the natural log of X.
  • 💡 The example of y = x natural log of X - 4x being a solution to the differential equation X + y - xy' = 0 demonstrates the application of derivatives in solving differential equations.
Q & A

  • What is the definition of a natural log?

    -The natural log, also known as the logarithm to the base e, is defined as the area under the curve of 1/T from 1 to T, and it is only defined for positive values of X.

  • Who was the natural logarithm named after?

    -The natural logarithm was named after Leonard Euler, a prominent mathematician.

  • What is the domain of the natural log function?

    -The domain of the natural log function is all positive real numbers, as it is undefined for zero or negative values.

  • What is the range of the natural log function?

    -The range of the natural log function is all real numbers, as it can take on any real value.

  • What are some properties of the natural log graph?

    -The graph of the natural log has an x-intercept at (1, 0), is always increasing, and is always concave down.

  • What is the relationship between the natural log and the number e?

    -The natural log and the number e are inverse functions of each other.

  • What is the derivative of the natural log of X with respect to X?

    -The derivative of the natural log of X with respect to X is 1/X.

  • How can you simplify the derivative of a natural log with an exponent?

    -You can simplify the derivative by bringing the exponent down and multiplying it by the derivative of the natural log of the base expression.

  • What is the process to find the derivative of a natural log using the fundamental theorem of calculus?

    -The process involves recognizing the natural log as the integral of 1/T from 1 to X, then applying the fundamental theorem of calculus which states that the derivative of an integral is the integrand evaluated at the upper limit of integration, multiplied by the derivative of the upper limit.

  • Can you provide an example of using the properties of logarithms to simplify a complex logarithmic expression?

    -Yes, for example, the natural log of x squared plus 3 can be rewritten using logarithmic properties as -1/3 times the natural log of (x squared plus 1).

  • What is the significance of the natural log in solving differential equations?

    -The natural log is significant in solving differential equations as it allows for the simplification of expressions and the application of logarithmic properties, making the process of finding solutions more manageable.

  • How does the chain rule apply to the derivative of a natural log of a function?

    -The chain rule is applied by taking the derivative of the inside function, multiplying it by the derivative of the outside function, and then dividing by the inside function itself, which in the case of a natural log is 1/U times U'.

  • What is the derivative of the natural log of a squared function?

    -The derivative of the natural log of a squared function, such as x squared, is twice the natural log of x, which simplifies to 2 times 1/x.

  • What is a common mistake when dealing with the derivative of a natural log with an exponent?

    -A common mistake is to incorrectly bring the exponent outside the natural log, which is not allowed unless it is an exponent on the base of the natural log itself.

Outlines
00:00
📚 Introduction to Natural Logs and Derivatives

This paragraph introduces the concept of natural logs, explaining that they are defined as the area under the curve of 1/T from 1 to T, and are only valid for positive values of X. The base of the natural log is the number E, named after Leonard Euler. The paragraph also discusses the properties of the natural log function, such as its graph, domain, range, and behavior (always increasing, concave down). It highlights the relationship between natural logs and the number E, stating that they are inverse functions. The speaker also mentions various properties of logarithms that were likely covered in precalculus, such as how to handle logarithms when they are multiplied, divided, or raised to a power. The paragraph sets the stage for learning how to take derivatives of natural logs.

05:03
🔍 Derivatives of Natural Logs and Logarithmic Properties

This paragraph delves into the process of finding derivatives of natural log functions. It starts by discussing the fundamental theorem of calculus and how it can be used to find the derivative of the natural log of X, which is 1/X. The speaker emphasizes the importance of understanding what this derivative means in terms of the slope of the natural log graph. Examples are provided to illustrate how to apply the chain rule and product rule when dealing with more complex natural log expressions. The paragraph also covers how to simplify expressions involving natural logs before taking derivatives, which can make the process easier. The speaker uses specific examples, such as the natural log of a squared term or a cubed term, to demonstrate these concepts. Additionally, the paragraph touches on the difference between taking the derivative of a natural log raised to a power versus a natural log of a power, highlighting the importance of understanding the underlying mathematical rules.

10:03
🧩 Advanced Derivatives and Differential Equations

The final paragraph covers more advanced topics related to derivatives of natural logs, including the use of substitution and the handling of differential equations. The speaker demonstrates how to use u-substitution to find the derivative of the natural log of the natural log of X, resulting in a simplified expression. The paragraph also discusses the difference between bringing down an exponent in front of a natural log versus a power on the entire natural log expression. The speaker provides an example of a differential equation and shows how to solve it using the derivatives of natural logs. The example involves finding the derivative of an expression involving both X and the natural log of X, and then verifying that the given function satisfies the differential equation. This paragraph concludes with a summary of the key takeaway: that the derivative of a natural log of U is U'/U, reinforcing the foundational concept introduced earlier in the script.

Mindmap
Keywords
💡Natural Logarithm
The natural logarithm, often denoted as ln(x), is the logarithm of a number to the base e, where e is an irrational constant approximately equal to 2.71828. It is the inverse function of the exponential function with base e. In the context of the video, the natural logarithm is introduced as an area under the curve of 1/T from 1 to X, and it is defined only for positive values of X. The video emphasizes its importance in various mathematical contexts such as half-life, exponential growth, and decay.
💡Derivative
A derivative in calculus represents the rate at which a function changes with respect to its variable. It is the slope of the tangent line to the function at a given point. The video focuses on how to take derivatives of natural logarithms, which is a fundamental concept in understanding the rate of change of logarithmic functions. For example, the derivative of the natural log of X is explained to be 1/X, illustrating the direct relationship between the slope of the graph and the reciprocal of the x-coordinate.
💡Euler's Number (e)
Euler's number, commonly denoted as 'e', is a fundamental mathematical constant that serves as the base of the natural logarithm. It is an irrational number and is approximately equal to 2.71828. The video mentions Euler as the person after whom the natural logarithm was named, highlighting its significance in mathematical analysis and the study of logarithms.
💡Properties of Logarithms
The properties of logarithms are a set of mathematical rules that simplify the process of working with logarithms. These include the product rule, quotient rule, and power rule. The video script discusses these properties, showing how they can be used to simplify complex logarithmic expressions, which is crucial when taking derivatives of such expressions.
💡Graph of Natural Log
The graph of the natural logarithm function is characterized by an x-intercept at (1, 0), and it is always increasing and concave down. The video describes the graph's features, such as its domain being all positive numbers and its range being all real numbers, with an asymptote on the y-axis. Understanding the graph is essential for visualizing the behavior of the natural logarithm function.
💡Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus links the concept of differentiation and integration, stating that the definite integral of a function can be found by finding the antiderivative of the function and then evaluating it at the limits of integration. The video uses this theorem to find the derivative of the natural logarithm function by differentiating its integral representation.
💡Chain Rule
The chain rule is a fundamental principle in calculus used to find the derivative of a composite function. It states that the derivative of a function composed of two functions is the derivative of the outer function times the derivative of the inner function. The video demonstrates the use of the chain rule in finding derivatives of functions involving natural logarithms, such as ln(2x) and ln(x^2 + 1).
💡Product Rule
The product rule is a derivative rule that applies to the product of two functions. It states that the derivative of the product is the derivative of the first function times the second function plus the first function times the derivative of the second function. The video uses the product rule to find the derivative of a function that is a product of x and the natural logarithm of x.
💡Quotient Rule
The quotient rule is a method in calculus for finding the derivative of a quotient of two functions. It is used when the function to be differentiated is in the form of one function divided by another. The video script mentions the quotient rule in the context of simplifying complex logarithmic expressions before differentiation.
💡Differential Equation
A differential equation is an equation that relates a function with its derivatives. The video concludes with an example of a differential equation involving the natural logarithm, demonstrating how to solve it by finding the derivative of a given function and showing that it satisfies the equation. This example illustrates the application of derivatives in solving practical problems.
Highlights

Introduction to natural logs and their derivatives.

Definition of natural log as the area under the curve of one over T DT.

Natural log is only defined for positive values of X.

The base of natural log is the number E, named after Leonard Euler.

Natural log and E are inverse functions of each other.

Properties of natural logs include an x-intercept at (1,0), always increasing, and concave down.

Domain of natural log is all positive numbers, range is all real numbers.

Natural log graph has an asymptote on the y-axis.

Natural log of 1 is 0 and natural log of E is 1.

How to separate logs in multiplication, division, or with a power of 1.

Derivative of natural log of X is 1 over X.

Slope of the natural log graph is the reciprocal of the x coordinate.

Derivative of natural log of 2x involves multiplying by the derivative of 2x.

Using u substitution to find the derivative of natural log of x squared plus 1.

Product rule applied to x times natural log of x.

Chain rule used for the derivative of natural log of the square root of x plus 1.

Natural log of the natural log of X involves u substitution with u = natural log of X.

Difference between bringing down an exponent in a natural log and a power on the whole natural log.

Example of solving a differential equation using derivatives of natural logs.

Transcripts

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Related Tags
Natural LogsDerivativesCalculusEulerExponential GrowthLogarithm PropertiesSlope InterpretationChain RuleProduct RuleDifferential Equations