Derivative of inverse tangent | Taking derivatives | Differential Calculus | Khan Academy

Khan Academy
1 May 201406:02
EducationalLearning
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TLDRThe video script discusses the process of finding the derivative of the inverse function of the tangent function, specifically focusing on the derivative of the inverse tangent function with respect to x. By setting y as the inverse tangent of x and applying the chain rule, the derivative is initially found in terms of y. Through the use of trigonometric identities and the Pythagorean identity, the expression is eventually rewritten in terms of x, resulting in the derivative of the inverse tangent function with respect to x being one over one plus the square of x.

Takeaways
  • 📚 The derivative of the tangent function with respect to x is the secant squared, which is equivalent to 1/(cosine of x)^2.
  • 🔄 To find the derivative of the inverse function, set y equal to the inverse tangent of x, which implies that the tangent of y equals x.
  • 📈 By applying the chain rule, the derivative of the tangent function with respect to y is the secant squared of y, and the derivative of y with respect to x can be found by multiplying these.
  • 🎓 The derivative of x with respect to x is simply 1, which is a constant.
  • 🔄 Solving for the derivative of y with respect to x involves multiplying both sides by the cosine of y squared.
  • 📊 The derivative of y with respect to x is initially expressed as a function of y, which needs to be rewritten as a function of x.
  • 🧠 To express the derivative as a function of x, introduce the tangent of y, knowing that tangent of y is equal to x.
  • 📐 Utilizing trigonometric identities, specifically the Pythagorean identity, helps to rewrite the expression in terms of sine and cosine functions.
  • 🔄 By dividing the numerator and denominator by the cosine squared of y, the expression simplifies to sine of y over cosine of y squared.
  • 🎉 The final form of the derivative of the inverse tangent function with respect to x is 1/(1 + x^2), which is a significant result of the process.
  • 📝 The process demonstrates the importance of understanding and applying chain rules, trigonometric identities, and the relationship between functions and their inverses in calculus.
Q & A

  • What is the derivative of the tangent function with respect to x?

    -The derivative of the tangent function with respect to x is the secant of x squared, which is equivalent to 1 divided by the cosine of x squared.

  • What is the inverse function of the tangent of x?

    -The inverse function of the tangent of x is the inverse tangent of x, often denoted as arctan(x) or tan^(-1)x.

  • How can we find the derivative of the inverse tangent function with respect to x?

    -We can find the derivative of the inverse tangent function by setting y equal to the inverse tangent of x, differentiating both sides of the equation with respect to x, and applying the chain rule.

  • What is the relationship between y and x when y is the inverse tangent of x?

    -When y is the inverse tangent of x, the relationship is such that the tangent of y is equal to x.

  • What is the derivative of the tangent function with respect to y?

    -The derivative of the tangent function with respect to y is the secant squared of y, which is the same as 1 divided by the cosine squared of y.

  • How does the Pythagorean identity help in finding the derivative of the inverse tangent function?

    -The Pythagorean identity, which states that sine squared y plus cosine squared y equals one, helps in expressing the derivative of the inverse tangent function in terms of x by allowing us to rewrite the cosine squared y term in terms of the tangent of y.

  • What is the final expression for the derivative of the inverse tangent function with respect to x?

    -The final expression for the derivative of the inverse tangent function with respect to x is 1 divided by 1 plus x squared, or written as 1/(1+x^2).

  • How can we rewrite the derivative of the inverse tangent function in terms of the tangent function?

    -We can rewrite the derivative in terms of the tangent function by using the relationship that tangent of y is equal to x, which allows us to express the cosine squared y term as 1 divided by 1 plus the square of the tangent of y, leading to the final expression of the derivative.

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

    -The chain rule is significant in this process as it allows us to differentiate the composite function, the inverse tangent, with respect to x by first finding the derivative of the tangent function with respect to y and then multiplying it by the derivative of y with respect to x.

  • Why is it important to express the derivative in terms of x rather than y?

    -Expressing the derivative in terms of x is important because it allows us to understand how the inverse tangent function changes with respect to changes in x, which is the independent variable in the context of the original problem.

Outlines
00:00
📚 Derivative of Inverse Tangent Function

This paragraph introduces the concept of finding the derivative of the inverse function of the tangent function, specifically focusing on the inverse tangent of x. The video encourages viewers to pause and attempt to solve the problem using techniques from previous videos. The process begins by setting y equal to the inverse tangent of x, which implies that the tangent of y equals x. By applying the chain rule and differentiating both sides with respect to x, the video derives the expression for the derivative of y with respect to x. The goal is to express this derivative in terms of x, not y, by using the known relationship that the tangent of y is equal to x. The paragraph concludes with a detailed explanation of the steps and mathematical manipulations involved in this process.

05:04
📈 Solving for Derivative using Trigonometric Identities

In this paragraph, the video continues the process of finding the derivative of the inverse tangent function by using trigonometric identities. The focus is on transforming the derived expression into a function of x. The video uses the Pythagorean identity to divide the expression by one, which simplifies the process. By dividing the numerator and denominator by the cosine squared of y, the video derives an expression involving sine and cosine functions. The final step is to substitute the tangent of y with x, leading to the conclusion that the derivative of y with respect to x is equal to one over one plus x squared. This paragraph provides a clear and detailed explanation of the mathematical steps and the final result of the derivative.

Mindmap
Keywords
💡derivative
The derivative is a fundamental concept in calculus that represents the rate of change of a function with respect to its variable. In the context of the video, the derivative is used to find the rate of change of the inverse function of the tangent function, specifically the inverse tangent function. The process involves applying the chain rule and understanding how the derivative behaves in relation to trigonometric functions.
💡tangent of x
The tangent of x is a trigonometric function that represents the ratio of the opposite side to the adjacent side in a right-angled triangle, or equivalently, the slope of the line that touches the curve at a given point. In the video, the tangent function is used as a basis for exploring its inverse function and the relationship between the derivative of the function and its inverse.
💡secant of x squared
The secant of x squared refers to the reciprocal of the cosine of x squared, which is a trigonometric identity used in the video to express the derivative of the tangent function. This concept is crucial in deriving the formula for the derivative of the inverse tangent function, as it relates to the rate of change of the tangent function.
💡inverse function
An inverse function is a mathematical concept where the roles of the independent variable (x) and the dependent variable (y) are swapped. In the video, the focus is on finding the derivative of the inverse function of the tangent function, which is the inverse tangent or arctangent function. Understanding inverse functions is key to solving this problem.
💡chain rule
The chain rule is a fundamental calculus technique used to find the derivative of a composite function. It states that the derivative of a function that is the result of other functions nested together is the derivative of the outer function times the derivative of the inner function. In the video, the chain rule is applied to find the derivative of the inverse tangent function with respect to x.
💡Pythagorean identity
The Pythagorean identity is a fundamental relationship in trigonometry that states that the square of the sine of an angle plus the square of the cosine of the same angle is equal to 1. This identity is used in the video to simplify and manipulate expressions involving the cosine squared term, ultimately leading to the derivation of the derivative of the inverse tangent function.
💡unit circle
The unit circle is a circle with a radius of 1 centered at the origin of a coordinate system. It is a key concept in trigonometry, as it provides a geometric interpretation of trigonometric functions. In the video, the unit circle is implicitly referenced through the Pythagorean identity, which is derived from the properties of the unit circle.
💡trigonometric identities
Trigonometric identities are equations that relate different trigonometric functions. They are used to simplify expressions and solve trigonometric equations. In the video, several identities are used, such as the Pythagorean identity, to transform and simplify the expression for the derivative of the inverse tangent function.
💡arctangent function
The arctangent function, also known as the inverse tangent function, is the inverse of the tangent function. It takes a ratio (the value of the tangent function) and returns an angle whose tangent is that ratio. In the video, the arctangent function is the main focus, with the goal of finding its derivative with respect to x.
💡rate of change
The rate of change is a fundamental 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 particular point. In the video, the rate of change is explored in the context of the tangent and inverse tangent functions, aiming to understand how the inverse function changes with respect to changes in x.
💡slope
The slope is a measure of the steepness of a line, representing the rate at which the y-value changes with respect to the x-value. In the context of the video, the slope is discussed in relation to the tangent function, which gives the slope of a line at a particular point on a curve. The derivative of the inverse tangent function is also related to the concept of slope, as it describes the rate of change of the inverse tangent with respect to x.
Highlights

The derivative of the tangent function with respect to x is the secant of x squared, which is equivalent to 1 divided by the cosine of x squared.

The goal is to find the derivative of the inverse function of the tangent function, specifically the inverse tangent of x.

By setting y equal to the inverse tangent of x, we establish that the tangent of y is equal to x.

Applying the chain rule to the derivative of both sides with respect to x allows us to find the derivative of y with respect to x.

The derivative of the tangent function with respect to y is the secant squared of y, or 1 divided by the cosine squared of y.

The derivative of x with respect to x is simply 1.

To solve for the derivative of y with respect to x, we multiply both sides by the cosine squared of y.

The derivative of y with respect to x is initially expressed as the cosine squared of y.

The aim is to express the derivative as a function of x, not y, by using the tangent of y, which is known to be equal to x.

Introducing trigonometric identities, specifically the Pythagorean identity, helps in expressing the derivative in terms of x.

The Pythagorean identity states that 1 is equal to the sum of sine squared y and cosine squared y.

Dividing the expression by 1, which is the sum of sine squared y and cosine squared y, does not change the value of the expression.

By dividing the numerator and denominator by the cosine squared of y, we achieve the form of sine divided by cosine squared.

The expression simplifies to sine of y over cosine of y, squared, which is equivalent to 1 over 1 plus the square of the tangent of y.

Since x is equal to the tangent of y, the derivative of y with respect to x can be expressed as 1 over 1 plus x squared.

The final result of the derivative of the inverse tangent function with respect to x is 1 divided by 1 plus x squared.

Transcripts

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