Derivative Trick That's NEVER Taught

BriTheMathGuy

11 Apr 201806:10

EducationalLearning

32 Likes 10 Comments

TLDRIn this educational video, Brian introduces a unique derivative trick derived from Pascal's Triangle to simplify the process of taking derivatives of product functions. He demonstrates the method by calculating the fourth and third derivatives of given functions in a single line, using coefficients from the triangle. This innovative approach not only impresses but also simplifies complex calculus problems, making it easier for students to grasp and apply.

Takeaways

🧩 Brian introduces a unique derivative trick that leverages Pascal's Triangle to simplify the process of finding derivatives of product functions.
📚 Pascal's Triangle is a mathematical pattern that can be used to determine coefficients for derivative calculations without memorization.
📉 The first row of Pascal's Triangle corresponds to the 0th derivative, the second row to the 1st derivative, and so on, up to the desired order of the derivative.
🔢 The trick involves using the coefficients from Pascal's Triangle to multiply the respective derivatives of the functions involved in the product.
📈 The example given demonstrates taking the fourth derivative of a product involving 'e^x' and a polynomial function.
🤔 For the fourth derivative, Brian uses the coefficients 1, 4, 6, 4, 1 from the fourth row of Pascal's Triangle to calculate the derivative.
📝 The process involves taking derivatives of the first function in one direction and the second function in the opposite direction, multiplying by the corresponding coefficients.
🔄 The derivative of 'e^x' is always 'e^x', which simplifies the process as it can be factored out and applied to each term.
📚 Another example is provided for taking the third derivative of a product involving sine and a polynomial function.
📉 For the third derivative, the coefficients from the third row of Pascal's Triangle, 1, 3, 3, 1, are used.
📝 The final expression for the third derivative is neatly written out, showcasing the application of the trick in a clear and concise manner.
👍 Brian encourages viewers to use this trick to impress others and to subscribe for more math-related content.

Q & A

What is the main topic of the video script?
-The main topic of the video script is a derivative trick involving Pascal's Triangle for simplifying the process of taking derivatives of products of functions.
Who is the presenter in the video script?
-The presenter in the video script is Brian.
What is the significance of Pascal's Triangle in the derivative trick?
-Pascal's Triangle is used to provide coefficients for the derivatives of the functions in the product, which simplifies the process of finding higher-order derivatives.
How does the derivative trick differ from the standard method of finding derivatives?
-The derivative trick uses the coefficients from Pascal's Triangle to take derivatives of both functions in a product separately and then combine them, which can be more efficient than the standard method for higher-order derivatives.
Can you provide an example of how the derivative trick is applied in the script?
-The script demonstrates the derivative trick by taking the fourth derivative of a product involving 'e^x' and a polynomial function, using coefficients from Pascal's Triangle to simplify the process.
What is the role of the coefficients from Pascal's Triangle in the derivative trick?
-The coefficients from Pascal's Triangle are used to multiply the derivatives of each function in the product, at different orders, to construct the final derivative expression.
How does the script illustrate the process of taking the derivative of 'e^x'?
-The script shows that the derivative of 'e^x' is simply 'e^x' itself, and this property is used repeatedly when taking higher-order derivatives of the function multiplied by 'e^x'.
What is the purpose of the example given for the third derivative in the script?
-The purpose of the example for the third derivative is to demonstrate the application of the derivative trick with Pascal's Triangle for a different order of derivative and a different function (sine and polynomial).
How does the script suggest one can use this derivative trick?
-The script suggests that one can use this derivative trick to impress friends, teachers, and to simplify the process of finding higher-order derivatives of products of functions.
What is the final call-to-action in the video script?
-The final call-to-action in the video script is to subscribe to the channel for more math tricks, and to like the video.

Outlines

00:00

📚 Derivative Trick with Pascal's Triangle

In this paragraph, Brian introduces a unique derivative trick that simplifies the process of differentiating products of functions. He demonstrates the power of this method by tackling a fourth derivative example in a single line. The trick is rooted in Pascal's Triangle, a mathematical pattern that can be easily constructed and used to determine coefficients for the derivatives of a product. Brian walks through the process of applying this trick to a function involving 'e to the x' and its derivatives, showing how to use the coefficients from Pascal's Triangle to perform the differentiation efficiently.

05:01

🔍 Applying the Derivative Trick to a Third Derivative

Brian continues to explore the derivative trick by applying it to find the third derivative of a product involving sine and cosine functions. He uses the coefficients from a specific row of Pascal's Triangle to guide the differentiation process. The explanation includes how to multiply the original functions by their respective derivatives and how to combine these terms to arrive at the final derivative. The example provided is a clear demonstration of how this method can simplify complex derivative calculations, making it an impressive tool for both students and educators.

Mindmap

Keywords

💡Derivative

A derivative in calculus is a measure of how a function changes as its input changes. It is the slope of the tangent line to the function at a particular point. In the video, the derivative is the main focus, as the presenter demonstrates a trick for taking the derivative of a product of two functions, which is a key concept in differential calculus.

💡Pascal's Triangle

Pascal's Triangle is a triangular array of numbers in which each number is the sum of the two numbers directly above it. It is used in the video as a mnemonic device for remembering the coefficients needed for the derivative trick. The presenter explains how the pattern in Pascal's Triangle corresponds to the coefficients of the terms when expanding the derivative of a product of functions.

💡Coefficients

In mathematics, coefficients are numerical factors that multiply variables in an algebraic expression. In the context of the video, the coefficients from Pascal's Triangle are used to multiply the derivatives of the functions involved in the product, which simplifies the process of finding higher-order derivatives.

💡Product of Functions

A product of functions refers to the result of multiplying two or more functions together. In the video, the presenter discusses a special case where the product rule is applied to find the derivative of a product of two functions, which is a common operation in calculus.

💡Fourth Derivative

The fourth derivative is the derivative of a function that has been taken four times. It represents the rate of change of the third derivative. The video uses the fourth derivative as an example to illustrate the derivative trick, showing how to calculate it efficiently using Pascal's Triangle.

💡Trigonometric Functions

Trigonometric functions are functions of an angle, including sine, cosine, and tangent. In the video, sine and cosine functions are used as examples of the first function in the product whose derivatives are being calculated, demonstrating the application of the derivative trick to these common mathematical functions.

💡Exponential Function

An exponential function is a function of the form f(x) = a * b^x, where a and b are constants, and b > 0, b ≠ 1. In the video, 'e to the x' is used as the second function in the product, showing how the derivative trick can be applied to both polynomial and exponential functions.

💡Polynomial

A polynomial is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, and multiplication. In the video, the presenter uses polynomials, specifically 'x to the fourth', to demonstrate the derivative trick with the second function in the product.

💡Derivative of e to the x

The derivative of the exponential function e^x is unique in that it is equal to the original function itself, e^x. This property is used in the video to simplify the process of taking derivatives of the exponential part of the product.

💡Third Derivative

The third derivative is the derivative of a function that has been taken three times. It is used in the video as another example to show the application of the derivative trick, further illustrating the method for finding higher-order derivatives of a product of functions.

💡Factoring Out

Factoring out is the process of taking a common factor from all terms in a mathematical expression. In the context of the video, the presenter factors out the exponential function 'e to the x' to simplify the expression when applying the derivative trick, making the calculation more manageable.

Highlights

Introduction of a derivative trick that 99% of people may not have seen before.

Demonstration of taking the fourth derivative of a product in one line using the trick.

The trick is derived from Pascal's triangle, a pattern that can be easily written down.

Explanation of how Pascal's triangle is used as coefficients for product derivatives.

Illustration of the process of using Pascal's triangle for the fourth derivative.

Description of how to apply the coefficients to derivatives of the first function.

Explanation of the process for applying the coefficients to derivatives of the second function in the opposite direction.

Example of using the trick to take the third derivative of a product.

Use of Pascal's triangle for the third derivative, requiring only a specific line from the triangle.

Step-by-step application of the rule for the third derivative using sine and cosine functions.

Application of the derivative trick to a polynomial function, x to the fourth plus 1.

Neatly writing out the derivative formula or factoring out terms for simplicity.

Encouragement to use the trick to impress friends, teachers, and for educational purposes.

Invitation to subscribe for more math tricks and educational content.

A call to action for viewers to like the video and engage with the content.

Closing remarks with a wish for the viewers to have a great day.

Transcripts

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Takeaways

Q & A

What is the main topic of the video script?

Who is the presenter in the video script?

What is the significance of Pascal's Triangle in the derivative trick?

How does the derivative trick differ from the standard method of finding derivatives?

Can you provide an example of how the derivative trick is applied in the script?

What is the role of the coefficients from Pascal's Triangle in the derivative trick?

How does the script illustrate the process of taking the derivative of 'e^x'?

What is the purpose of the example given for the third derivative in the script?

How does the script suggest one can use this derivative trick?

What is the final call-to-action in the video script?