How To Find The Derivative of Sin^2(x), Sin(2x), Sin^2(2x), Tan3x, & Cos4x

The Organic Chemistry Tutor
21 Jul 202005:23
EducationalLearning
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TLDRThe video script offers a comprehensive guide on calculating derivatives of common trigonometric functions, essential for homework or test preparation. It explains the process step-by-step, starting with basic functions like the derivative of sine(2x) and tangent(3x), and progressing to more complex examples such as the derivative of sine squared(2x) and sine(tangent(x squared)). The method involves finding the derivative of the outer function, handling the inner function, and applying the chain rule and properties of trigonometric identities. The script is an informative resource for those looking to enhance their understanding of derivatives in trigonometry.

Takeaways
  • 📚 The derivative of sin(2x) is 2cos(2x), found by applying the chain rule to the sine function and differentiating 2x.
  • 📈 The derivative of tan(3x) is 3sec^2(3x), using the derivative of the tangent function (sec^2) and the derivative of the inner function (3x).
  • 🌟 The derivative of cos(4x) is -4sin(4x), keeping the angle the same for the cosine function and applying the derivative to 4x.
  • 🔍 To find the derivative of (sin(2x))^2, rewrite the expression as 2sin(2x) - 1 and apply the chain rule to the exponent, resulting in 4sin(2x)cos(2x).
  • 📝 For the derivative of sin^2(x), simplify the expression by moving the exponent to the outside and applying the chain rule, leading to 2sin(x)cos(x).
  • 🤔 The derivative of sin(tan(x^2)) involves differentiating both the sine and tangent functions, resulting in 2xsec^2(x^2)cos(x^2).
  • 📊 The process of finding derivatives of trigonometric functions often involves the chain rule and understanding the basic derivatives of trigonometric functions like sine and cosine.
  • 🔧 The use of the chain rule is crucial for differentiating composite functions, which is a common scenario in calculus problems involving trigonometric functions.
  • 🧩 When dealing with trigonometric functions, it's important to keep track of the angle transformations and the corresponding derivatives.
  • 📐 The derivatives of trigonometric functions are fundamental in solving calculus problems and can be applied in various areas of mathematics and physics.
Q & A

  • What is the derivative of sine of 2x?

    -The derivative of sine of 2x is 2 cosine of 2x. This is found by first recognizing that the derivative of sine is cosine, and then applying the chain rule to account for the 2x, resulting in 2 multiplied by the derivative of cosine at 2x, which is 2 cosine of 2x.

  • How do you find the derivative of tangent of 3x?

    -The derivative of tangent of 3x is 3 secant squared of 3x. This is calculated by first identifying the derivative of tangent as secant squared, and then using the chain rule for the 3x, which gives us 3 multiplied by the derivative of secant squared at 3x.

  • What is the derivative of cosine of 4x?

    -The derivative of cosine of 4x is negative 4 sine of 4x. This is derived by recognizing that the derivative of cosine is negative sine, and then applying the chain rule to the 4x, resulting in -4 times the derivative of sine at 4x, which is -4 sine of 4x.

  • How does the video suggest simplifying the derivative of sine squared of 2x?

    -The video suggests simplifying the derivative of sine squared of 2x by first rewriting the expression as if sine of 2x were raised to the power of 2. This allows you to treat the 2 as an exponent, which when differentiated using the power rule becomes 2 times sine of 2x. Then, you subtract the exponent by 1 and multiply by the derivative of sine, which is cosine, resulting in 2 cosine of 2x times sine of 2x.

  • What is the process for finding the derivative of sine squared of x?

    -To find the derivative of sine squared of x, you first rewrite the expression as if sine of x were raised to the power of 2. Then, you apply the power rule to move the exponent to the front, subtract the exponent by 1, and finally take the derivative of the inside function, which results in 2 sine of x times cosine of x.

  • How do you find the derivative of sine of tangent of x squared?

    -To find the derivative of sine of tangent of x squared, you first differentiate sine to get cosine, and then differentiate tangent to get secant squared. The angle for the secant squared derivative is x squared. Finally, you multiply the derivative of sine, which is cosine, by the derivative of x squared, which is 2x, and apply the secant squared to x squared, resulting in 2x cosine of tangent of x squared times secant squared of x squared.

  • What is the chain rule used for in differentiation?

    -The chain rule is used in differentiation when you need to differentiate a function that is composed of one function nested inside another. It allows you to break down the process by differentiating the outer function first, then the inner function, and finally combining the results to find the derivative of the entire composite function.

  • What is the power rule used for in differentiation?

    -The power rule is used in differentiation when you need to differentiate a function that involves an exponent. It states that the derivative of x to the power of n is n times x to the power of n minus 1. This rule simplifies the process of differentiating expressions that involve exponents.

  • Why is it beneficial to rewrite expressions when differentiating?

    -Rewriting expressions can make the differentiation process easier and more straightforward. By rearranging terms or factoring out common elements, you can simplify complex expressions and apply differentiation rules more effectively, leading to a clearer path to the final derivative.

  • What is the significance of the double angle formula in the context of differentiation?

    -The double angle formula is significant in differentiation as it allows you to express the derivative of a function involving a squared sine in terms of a product of the original function and another trigonometric function. This simplifies the expression and makes it easier to work with, especially when dealing with composite functions or more complex trigonometric expressions.

  • How does the video demonstrate the use of trigonometric identities in differentiation?

    -The video demonstrates the use of trigonometric identities, such as the double angle formula, in differentiation by showing how they can simplify the process. For example, after differentiating the sine squared expression, the video uses the double angle formula to express the result in a more recognizable form, which can be helpful for further analysis or for checking the solution.

Outlines
00:00
📚 Introduction to Derivatives of Trigonometric Functions

This paragraph introduces the topic of finding derivatives of common trigonometric functions. It begins with an example problem, showing how to find the derivative of sine of 2x. The process involves first identifying the derivative of the base function, which is cosine for sine, and then applying the chain rule to the argument (2x), resulting in the final answer of 2 cosine 2x. The paragraph continues with additional examples, such as finding the derivative of tangent 3x and cosine 4x, using similar techniques. It also touches on more complex problems, like finding the derivative of sine squared of 2x, and provides a method for solving such problems by rewriting the expression and applying the power rule.

05:01
🔢 Solving Derivatives of Composite Trigonometric Functions

This paragraph delves into solving derivatives of composite trigonometric functions. It starts by addressing the problem of finding the derivative of sine of tangent x squared. The approach involves differentiating the outer function (sine) and the inner function (tangent x squared) separately. The derivative of tangent is secant squared, and the derivative of x squared is 2x. The paragraph then combines these results with the chain rule to arrive at the final answer, which is 2x cosine of tangent x squared times secant squared x squared. The discussion serves as a comprehensive guide on how to tackle more complex trigonometric derivatives by breaking them down into manageable parts.

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, it is used to find the rate at which trigonometric functions change, which is crucial for solving related homework problems or test questions. For instance, the derivative of the sine function is the cosine function, as demonstrated when finding the derivative of sine of 2x, which results in 2 cosine of 2x.
💡trigonometric functions
Trigonometric functions, such as sine, cosine, and tangent, are mathematical functions that relate angles to real numbers. These functions are widely used in mathematics, physics, and engineering to model periodic phenomena. In the video, the focus is on finding the derivatives of these functions, which is essential for understanding their behavior and rates of change.
💡chain rule
The chain rule is a crucial technique in calculus used to find the derivative of composite functions, which are functions composed of other functions. It involves differentiating the outer function and then multiplying by the derivative of the inner function. The video demonstrates the chain rule by differentiating the sine of 2x, where the outer function is sine and the inner function is 2x.
💡angle
In trigonometry, the angle is a fundamental concept that represents the measure of rotation between two intersecting lines or the space between two intersecting lines in a plane. The angle is used in trigonometric functions to determine the values of these functions at different points. In the video, angles are used as inputs for the trigonometric functions when finding their derivatives.
💡secant
Secant is a trigonometric function that is the reciprocal of the cosine function. It is used in advanced trigonometry and calculus, particularly when dealing with the derivatives of tangent functions. The secant function becomes important in the video when finding the derivative of tangent, which is the secant squared.
💡exponents
Exponents are mathematical expressions that indicate repeated multiplication of a base by itself a certain number of times. They are used in various mathematical operations, including differentiation when dealing with functions raised to powers. In the video, exponents are crucial when finding the derivatives of functions like sine squared of 2x, where the exponent rules are applied to simplify the differentiation process.
💡differentiation
Differentiation is the process of finding the derivative of a function, which gives the rate of change or the slope of the function at any point. It is a fundamental operation in calculus and is essential for analyzing the behavior of functions. The video focuses on differentiation as it applies to trigonometric functions, showing how to find their derivatives to solve mathematical problems.
💡sine
Sine is one of the six trigonometric functions, which relates the ratio of the lengths of the sides of a right triangle to the angles within the triangle. It is a periodic function that oscillates between -1 and 1 and is used in various mathematical and real-world applications. In the video, the sine function and its derivative, cosine, are discussed in the context of differentiating trigonometric expressions.
💡cosine
Cosine is another fundamental trigonometric function that, like sine, relates the angles of a right triangle to the lengths of its sides. It is also a periodic function with values ranging from -1 to 1 and is used in numerous mathematical and scientific contexts. In the video, the cosine function is highlighted as the derivative of the sine function and is involved in several differentiation examples.
💡tangent
Tangent is a trigonometric function that represents the ratio of the length of the opposite side to the length of the adjacent side in a right triangle. It is used in various mathematical and real-world applications, particularly in the study of angles and slopes. The derivative of the tangent function is secant squared, which is emphasized in the video when differentiating trigonometric expressions involving tangent.
💡power rule
The power rule is a fundamental rule in calculus that allows for the differentiation of functions raised to a power. It states that the derivative of a function x raised to the power n is n times the function raised to the power (n-1). This rule is applied in the video when differentiating expressions like sine squared of 2x, simplifying the process by treating the exponent as a separate factor.
💡double angle formula
The double angle formula is a trigonometric identity that expresses a trigonometric function of twice an angle in terms of the function of the original angle. It is a useful tool in simplifying and solving trigonometric equations. In the video, the double angle formula is mentioned when differentiating sine squared of x, where 2sin(x)cos(x) is used to express the result.
Highlights

The video discusses the process of finding derivatives of common trigonometric functions.

Derivative of sine of 2x is found by first recognizing the derivative of sine is cosine.

For the sine of 2x, the angle inside the sine function remains the same as that of the cosine function.

The derivative of 2x is 2, which is applied when finding the derivative of sine of 2x.

The final answer for the derivative of sine of 2x is 2 cosine of 2x.

The derivative of tangent of 3x involves recognizing the derivative of tangent as secant squared.

The angle for the derivative of 3x is 3, leading to the answer 3 secant squared of 3x.

The derivative of cosine is negative sine, which is used to find the derivative of cosine of 4x.

The derivative of 4x is 4, and when combined with the derivative of cosine, the final answer is -4 sine of 4x.

To find the derivative of sine squared of 2x, the problem is rewritten to focus on the exponents.

The derivative of x squared becomes 2x to the first power when differentiating.

The final answer for the derivative of sine squared of 2x is 4 sine of 2x multiplied by cosine of 2x.

For the derivative of sine squared of x, the process involves moving the exponent to the outside and simplifying.

The derivative of 2x is 2, and when combined with the derivative of sine and cosine, the final answer is 2 sine of x times cosine of x.

The derivative of sine of tangent of x squared involves differentiating both the sine and tangent functions.

The derivative of x squared is 2x, and when applied to the tangent function, the final answer is 2x times cosine of tangent of x squared times secant squared of x squared.

Transcripts

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