Calculus - Find the derivative of inverse trigonometric functions

MySecretMathTutor
27 Aug 201710:12
EducationalLearning
32 Likes 10 Comments

TLDRThis video from MySecretMathTutor dives into the intricacies of differentiating inverse trigonometric functions, which may appear daunting but are essential for advanced calculus. The presenter begins with an example involving the derivative of a function composed of a polynomial and the inverse cotangent function, emphasizing the unique form these derivatives take. The video then outlines the derivative rules for six inverse trigonometric functions, highlighting that memorizing the rules for arcsine, arctangent, and arcsecant is key, as the others follow similar patterns but may include a negative sign. Practical examples are provided to illustrate the application of these rules, including the use of the chain rule and product rule in conjunction with the newly introduced derivatives. The presenter also advises viewers to check their math books for any variations in definitions, as these can affect the derivative rules. The video concludes with a reminder of the importance of these derivatives not just for differentiation but also for integration in future studies.

Takeaways
  • 📚 Start by understanding the derivatives of inverse trigonometric functions, which include inverse tangent (arctangent), inverse sine (arcsine), and inverse secant (arcsecant).
  • 🔢 The derivative of a function like y = 3x^2 + cot^-1(x) involves first taking the derivative of the polynomial part (3x^2 gives 6x) and then addressing the inverse trigonometric part.
  • ⚠️ For the derivative of inverse cotangent, the result is negative one divided by 1 + x squared, which simplifies to (6x - 1) / (1 + x^2).
  • 📉 Notably, the derivatives of inverse trigonometric functions do not contain any trigonometric functions after simplification, instead resulting in fractions with x terms and square roots.
  • 📋 There are six main rules for the derivatives of inverse trigonometric functions, but only three core rules need to be memorized: for inverse sine, inverse tangent, and inverse secant.
  • 🔁 The other three rules (inverse cosine, inverse cosecant, and inverse cotangent) are similar to the first three but may include a negative sign.
  • 🧮 The derivatives of inverse sine, inverse tangent, and inverse secant are given by 1/√(1-x^2), 1/(1+x^2), and 1/(x√(x^2-1)), respectively.
  • 📘 Be aware that the definitions and rules used in the video might differ from those in your math book, so it's important to check your textbook for the correct definitions and any restrictions or absolute values.
  • 📌 When applying the derivatives of inverse trigonometric functions, use the chain rule, product rule, and quotient rule as needed, just as with regular derivatives.
  • 📐 For example, the derivative of sin^-1(4x^2) is found by applying the chain rule to get (8x)/(√(1-(4x^2)^2)) or (8x)/(√(1-16x^4))
  • 📈 The derivative of e^x * tan^-1(x) uses the product rule and results in e^x * (1/(1+x^2)) + e^x * (0), simplifying to e^x / (1 + x^2).
  • 📒 Memorizing the derivatives of inverse trigonometric functions is crucial for recognizing them in integral calculus and for successfully completing homework assignments.
Q & A

  • What are the main topics covered in the video?

    -The video primarily covers the process of finding derivatives of inverse trigonometric functions, including inverse tangent, inverse sine (arcsine), and inverse cotangent.

  • What is the derivative of 3x^2?

    -The derivative of 3x^2 is 6x.

  • How is the derivative of the inverse cotangent function expressed?

    -The derivative of the inverse cotangent function is expressed as negative one divided by one plus x squared, which can be simplified to -(1)/(1 + x^2).

  • What are the three main rules one should remember for derivatives of inverse trigonometric functions?

    -The three main rules to remember are for the derivatives of inverse sine, inverse tangent, and inverse secant, as the others can be derived from these with a possible change in sign.

  • What does the derivative of the inverse sine function look like?

    -The derivative of the inverse sine function is one divided by the square root of one minus x squared, or (1)/(√(1 - x^2)).

  • What is the derivative of the inverse tangent function?

    -The derivative of the inverse tangent function is one divided by one plus x squared, or (1)/(1 + x^2).

  • What is the derivative of the inverse secant function?

    -The derivative of the inverse secant function is one divided by x, multiplied by the square root of x squared minus one, or (1/x) * √(x^2 - 1).

  • Why might the definitions of the inverse trigonometric functions in a math book differ from the ones in the video?

    -The definitions might differ due to different setups or parameterizations used in the math book, which could include restrictions or the use of absolute values.

  • What is the importance of knowing the derivative rules for inverse trigonometric functions when doing integrals?

    -Knowing these derivative rules is important because they will be needed in reverse when solving integrals, which is a common operation in calculus.

  • What is the derivative of sin^-1(4x^2) using the chain rule?

    -The derivative is 8x divided by the square root of one minus four times x squared, or (8x)/(√(1 - 4x^2)).

  • What is the product rule used for in the video?

    -The product rule is used to find the derivative of a function that is the product of two other functions, such as e^x multiplied by tan^-1(x).

  • What is the derivative of csc^-1(e^x)?

    -The derivative is negative one divided by the square root of e^(2x) minus one, or -(1)/(√(e^(2x) - 1)).

Outlines
00:00
📚 Introduction to Derivatives of Inverse Trigonometric Functions

This paragraph introduces the topic of finding derivatives of inverse trigonometric functions, which include inverse tangent (cot^-1), inverse sine (sin^-1), and others. The presenter uses the example y = 3x^2 + cot^-1(x) to demonstrate the process. It's noted that these derivatives do not contain trigonometric functions after simplification and instead result in fractions. The paragraph also emphasizes the importance of memorizing the derivatives of inverse sine, tangent, and secant, as they are fundamental to understanding the other three.

05:05
🔍 Deriving Specific Inverse Trigonometric Functions

The paragraph delves into the process of finding derivatives of specific inverse trigonometric functions. It demonstrates the derivative of sin^-1(4x^2) using the chain rule and the derivative formula for inverse sine. The presenter also covers the derivative of a function involving e^x and tan^-1(x), applying the product rule and noting the derivative of tan^-1(x). The paragraph further explains the derivative of csc^-1(e^x), highlighting the presence of two x terms in the derivative formula. The importance of recognizing these derivatives for future integrals is stressed, and viewers are advised to verify their math book's definitions to avoid confusion.

10:05
📝 Conclusion and Further Resources

The final paragraph concludes the discussion on derivatives of inverse trigonometric functions. It encourages viewers to memorize the rules for derivatives as they will be important for future studies, particularly when dealing with integrals. The presenter also advises viewers to check their math books for any variations in definitions and to subscribe to the YouTube channel for more educational content. The paragraph ends with a prompt to visit MySecretMathTutor.com for additional resources.

Mindmap
Keywords
💡Derivative
A derivative in calculus represents the rate at which a function changes with respect to a variable. It is a fundamental concept used to analyze the behavior of functions and is central to the video's theme of finding derivatives of inverse trigonometric functions. For example, the script discusses taking the derivative of 3x^2, which results in 6x.
💡Inverse Trigonometric Functions
Inverse trigonometric functions are the reverse of the standard trigonometric functions and include arcsine, arccosine, arctangent, etc. They are used to find the angles when the lengths of the sides of a triangle are known. In the video, the focus is on finding the derivatives of these functions, such as the inverse tangent or cotangent.
💡Cotangent
Cotangent is a trigonometric function that is the reciprocal of the tangent function, defined as cot(x) = 1/tan(x). In the context of the video, the derivative of the inverse cotangent function is discussed, which is an important part of understanding how to derive inverse trigonometric functions.
💡Chain Rule
The chain rule is a fundamental theorem in calculus used to compute the derivative of a composite function. It states that the derivative of a function composed of two functions is the product of the derivative of the outer function and the derivative of the inner function. The video uses the chain rule to find derivatives of functions like sin^-1(4x^2).
💡Product Rule
The product rule is a formula used to find the derivative of a product of two functions. It states that the derivative of the product is the first function times the derivative of the second function plus the second function times the derivative of the first function. The video uses the product rule in the example of e^x multiplied by tan^-1(x).
💡Inverse Sine (arcsin)
The inverse sine function, often denoted as arcsin or sin^-1, is an inverse trigonometric function that gives the angle whose sine is a given number. The video emphasizes the derivative of the inverse sine function, which is a key part of the content.
💡Inverse Tangent (arctan)
The inverse tangent function, denoted as arctan or tan^-1, finds the angle whose tangent is a given number. The video discusses the derivative of the inverse tangent and how it is used in calculus problems involving this function.
💡Inverse Secant (arcsec)
The inverse secant function, written as arcsec or sec^-1, is another inverse trigonometric function that finds the angle from the reciprocal of the cosine function. The video mentions the derivative of the inverse secant, which is part of the broader set of rules for derivatives of inverse trigonometric functions.
💡Square Root
A square root is a value that, when multiplied by itself, gives the original number. It is a mathematical concept that appears in the derivatives of inverse trigonometric functions, as seen in the video when discussing the derivative of arcsin(4x^2), which involves a square root in its expression.
💡Trigonometric Functions
Trigonometric functions are mathematical functions that relate angles to the sides of a right triangle. They include sine, cosine, and tangent, among others. The video's theme revolves around the derivatives of their inverse functions, which are used to solve more complex calculus problems.
💡Quotient Rule
The quotient rule is used to find the derivative of a quotient of two functions. It is another fundamental rule in calculus that complements the product and chain rules. Although not explicitly used in the provided examples, the video script implies its importance in the broader context of derivative calculations.
Highlights

The video focuses on finding derivatives of inverse trigonometric functions, which can be complex and different from regular derivatives.

An example problem, y = 3x^2 + cot^-1(x), is used to illustrate the process of finding derivatives of such functions.

Derivatives of inverse trigonometric functions do not contain trigonometric functions after simplification, turning into fractions instead.

There are six inverse trigonometric functions with six corresponding rules, but only three essential rules to remember: inverse sine, inverse tangent, and inverse secant.

The other three rules are variations of the first three, often with a negative sign difference, and are connected through their cofunctions.

Derivatives of inverse sine, tangent, and secant are given as specific fractions involving x squared or square roots.

Different math books may define these functions differently, so it's important to check the definitions against the rules provided in the video.

The video demonstrates the use of the chain rule, product rule, and quotient rule in conjunction with the new derivative rules.

An example is shown where the derivative of sin^-1(4x^2) is calculated, emphasizing the absence of trigonometric functions in the final derivative.

The derivative of tan^-1(x) is highlighted as a simple fraction, 1 / (1 + x^2), without involving square roots.

The product rule is applied to the function e^x * tan^-1(x), showcasing the multiplication of the function by its derivative.

The derivative of csc^-1(e^x) is calculated, noting the presence of two x terms in the derivative formula.

Common factors, such as e^x, are simplified in the derivative to clean up the expression.

The importance of memorizing these derivative rules is emphasized for future recognition in integral calculus.

The video encourages viewers to like, subscribe, and visit MySecretMathTutor.com for more educational content.

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