Worked example: Derivative of ln(Ãx) using the chain rule | AP Calculus AB | Khan Academy

Khan Academy
22 Jul 201606:44
EducationalLearning
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TLDRThe video script explains the process of finding the derivative of the function f(x) = ln(√x) using the concept of composite functions and the chain rule. It breaks down the function into two simpler functions, u(x) = √x and v(x) = ln(x), and applies the chain rule to find the derivative, f'(x) = (1/√x) * (1/(2√x)). The explanation is clear, methodical, and employs visual aids to enhance understanding, making it an effective educational resource for calculus students.

Takeaways
  • 📚 The main topic is finding the derivative of the function f(x) = ln(√x).
  • 🔍 The function f(x) can be viewed as a composition of two simpler functions.
  • 🌟 The first function, u(x), represents taking the square root of x (u(x) = √x).
  • 🌟 The second function, v(x), represents taking the natural log of x (v(x) = ln(x))
  • 📈 To find the derivative of f(x), the chain rule is applied, which involves the derivatives of both u(x) and v(x).
  • 🔢 The derivative of u(x) is found using the power rule, resulting in u'(x) = 1/(2√x).
  • 🔢 The derivative of v(x) is a standard result: v'(x) = 1/x.
  • 🔧 Applying the chain rule, v'(u(x)) is calculated by substituting u(x) into v'(x), yielding v'(u(x)) = 1/(√x).
  • 📊 The final derivative of f(x) is determined by multiplying u'(x) and v'(u(x)), resulting in f'(x) = 1/(2x).
  • 💡 Recognizing composite functions and applying the chain rule is crucial for solving such calculus problems.
  • 🚀 With practice, the process of finding derivatives of composite functions becomes more intuitive and requires less detailed step-by-step calculation.
Q & A

  • What is the main topic of the video?

    -The main topic of the video is finding the derivative of the function f(x), which is defined as the natural log of the square root of x, by recognizing it as a composition of two functions and applying the chain rule.

  • How is f(x) represented as a composition of two functions?

    -f(x) is represented as a composition of two functions where the first function, u(x), takes the square root of x, and the second function, v(x), takes the natural log of its input. Thus, f(x) can be expressed as v(u(x)).

  • What is the role of the chain rule in this process?

    -The chain rule is crucial in finding the derivative of the composite function f(x). It allows us to break down the derivative of f into the product of the derivative of the outer function (v') evaluated at the inner function (u(x)), and the derivative of the inner function (u') with respect to x.

  • What are the individual functions u(x) and v(x) in this context?

    -In this context, u(x) is the function that takes the square root of x, and v(x) is the function that takes the natural log of its input.

  • How is the derivative of u(x) calculated?

    -The derivative of u(x), which is the square root of x, is calculated using the power rule. It is derived as 1/2 * x^(-1), which simplifies to 1/(2*sqrt(x)).

  • What is the derivative of the natural log function, and how is it used in this video?

    -The derivative of the natural log function is 1/x. In the video, it is used as v'(x), and when evaluated at u(x) (the square root of x), it becomes 1/sqrt(x).

  • What is the final expression for the derivative of f(x)?

    -The final expression for the derivative of f(x) is (1/sqrt(x)) * (1/(2*sqrt(x))) which simplifies to 1/(2*x).

  • How does the video emphasize the importance of recognizing composite functions?

    -The video emphasizes that recognizing composite functions is key to understanding and applying the chain rule effectively. It helps in breaking down complex functions into simpler components, making the process of differentiation more manageable and intuitive.

  • What is the significance of the diagram used in the video?

    -The diagram used in the video is a visual representation of the composite function and its components. It aids in visualizing the process of inputting x into the functions, step by step, and helps to clarify the application of the chain rule.

  • How does the video suggest practicing for understanding the chain rule?

    -The video suggests that with practice, one can eventually recognize composite functions and apply the chain rule without having to write out all the steps. It encourages the development of a mental 'muscle' for identifying and working with composite functions and their derivatives.

  • What is the role of the Wolfram|Alpha Derivative Calculator in learning derivatives?

    -The Wolfram|Alpha Derivative Calculator is a tool that can help users understand and compute derivatives. It provides expert-level knowledge and capabilities, catering to a wide range of users across different professions and education levels. It can solve for first, second, and third derivatives, derivatives at a point, and partial derivatives, offering a comprehensive resource for learning about and applying calculus concepts.

Outlines
00:00
📚 Introduction to Derivative of Composite Functions

This paragraph introduces the concept of finding the derivative of a function, specifically focusing on a function f(x) which is a composition of two other functions. The function f(x) is defined as the natural log of the square root of x. The voiceover explains the process of breaking down the composite function into simpler functions, u(x) and v(x), where u(x) represents the square root of x and v(x) represents the natural log of x. The goal is to use the chain rule to find the derivative of f(x), which involves understanding the individual derivatives of u(x) and v(x), and how they relate to each other in the composition.

05:02
🧮 Applying the Chain Rule for Derivatives

This paragraph delves into the application of the chain rule for derivatives. It explains how to evaluate the derivative of the composite function f(x) by multiplying the derivative of the outer function (v) with respect to the inner function (u), by the derivative of the inner function (u) with respect to x. The paragraph provides a step-by-step breakdown of the calculations, including the derivative of the square root of x (u(x)) as 1/(2*sqrt(x)), and the derivative of the natural log of x (v(x)) as 1/x. The final derivative of f(x) is simplified to 1/(2*sqrt(x)) by applying algebraic rules and understanding the relationship between the functions in the composition.

Mindmap
Keywords
💡Natural Logarithm
The natural logarithm, often denoted as ln(x), is a logarithm to the base e (where e is an irrational number approximately equal to 2.71828). It is a fundamental concept in calculus and is used to model natural growth and decay processes. In the video, the natural logarithm is applied to the square root of x to form the function f(x) = ln(sqrt(x)), which is a key component in the process of finding the derivative of the function.
💡Square Root
The square root of a number x, denoted as √x, is a value that, when multiplied by itself, gives the original number x. It is a basic mathematical operation with applications in various fields. In the context of the video, taking the square root of x is the first step in defining the function f(x), which is then followed by taking the natural logarithm of the result.
💡Derivative
The derivative of a function represents the rate of change or the slope of the function at any given point. It is a fundamental concept in calculus used to analyze the behavior of functions. In the video, the main objective is to find the derivative of the function f(x), which involves understanding the composition of the function and applying the chain rule.
💡Composition of Functions
A composition of functions is a function that is made by applying one function to the result of another function. It is a way to combine different functions to create new ones with unique properties. In the video, the function f(x) is viewed as a composition of two simpler functions: u(x) = √x and v(x) = ln(x), which helps in understanding the process of finding the derivative of f(x).
💡Chain Rule
The chain rule is a fundamental rule in calculus that is used to find the derivative of a composite function. It states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. In the video, the chain rule is crucial for finding the derivative of f(x) = ln(sqrt(x)), as it allows us to break down the composite function into simpler parts and apply the derivatives accordingly.
💡Power Rule
The power rule is a basic differentiation rule in calculus that states that the derivative of x raised to the power n (where n is a constant) is n times x to the power of (n-1). In the video, the power rule is applied to find the derivative of the inner function u(x) = √x, resulting in u'(x) = 1/2 * x^(-1/2) or 1/(2√x).
💡Derivative of Natural Log
The derivative of the natural logarithm function, ln(x), with respect to x is 1/x. This means that for any value of x, the slope of the natural logarithm function at that point is the reciprocal of x. In the video, this fact is used to determine the derivative of the outer function v(x) = ln(x) when applied to the inner function's output.
💡Algebraic Simplification
Algebraic simplification involves using the rules of algebra to make expressions easier to understand or calculate. In the video, algebraic simplification is used to simplify the expression for the derivative of the composite function f(x) = ln(sqrt(x)), resulting in a final answer of (1/2) / √x.
💡Function Composition
Function composition is the process of applying one function to the result of another. It is often denoted as (f ∘ g)(x) = f(g(x)), where f is the outer function and g is the inner function. In the video, function composition is used to express f(x) as the composition of u(x) = √x and v(x) = ln(x), which simplifies the process of finding the derivative of f(x).
💡Rate of Change
The rate of change is a concept in calculus that describes how a quantity changes in response to changes in another quantity. It is the core idea behind derivatives, which measure the rate of change of a function at a specific point. In the video, the rate of change is the central theme, as the derivative of the function f(x) represents the rate at which the natural logarithm of the square root of x changes with respect to x.
💡Calculus
Calculus is a branch of mathematics that deals with rates of change and accumulation. It has two main branches: differential calculus, which deals with instantaneous rates of change and slopes of curves, and integral calculus, which deals with accumulation and areas under curves. In the video, calculus is the subject matter, with a focus on differential calculus as it pertains to finding derivatives of functions.
Highlights

The main goal of the video is to find the derivative of the function f(x) = ln(√x).

f(x) can be viewed as a composition of two functions, which is a key concept in understanding the chain rule.

The first function, u(x), takes the square root of x, which is represented as u(x) = √x.

The second function, v(x), takes the natural log of its input, represented as v(x) = ln(x).

The composition of functions is expressed as f(x) = v(u(x)), which simplifies to f(x) = ln(√x).

The chain rule is crucial for finding the derivative of composite functions and is expressed as f'(x) = v'(u(x)) * u'(x).

The derivative of u(x) = √x is found using the power rule and is expressed as u'(x) = 1/(2√x).

The derivative of v(x) = ln(x) is a standard result from calculus, which is v'(x) = 1/x.

To find v'(u(x)), we replace x with u(x) = √x in the expression for v'(x), resulting in v'(u(x)) = 1/√x.

The final expression for the derivative f'(x) is a product of v'(u(x)) and u'(x), which simplifies to f'(x) = (1/√x) * (1/(2√x)).

The simplified derivative f'(x) is 1/(2x), which is the main result of the video.

The video emphasizes the importance of recognizing composite functions and applying the chain rule effectively.

The process of finding the derivative involves understanding the individual functions and their roles in the composition.

The video uses a step-by-step approach to derive the chain rule, making it easier for viewers to understand and apply.

The method demonstrated in the video can be generalized to find derivatives of other composite functions in calculus.

The video provides a visual representation of the functions and their composition, aiding in the comprehension of the chain rule.

The video encourages practice and familiarity with the chain rule to streamline the process of finding derivatives.

The video serves as a practical guide for students learning about derivatives and the chain rule in their calculus studies.

Transcripts

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