Derivative of Sine and Cosine Functions | Calculus

The Organic Chemistry Tutor
23 Feb 201810:30
EducationalLearning
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TLDRThis lesson delves into the derivatives of sine and cosine functions, highlighting the fundamental relationships between these trigonometric functions and their derivatives. The derivative of sine x is shown to be cosine x, and the derivative of cosine x is negative sine x. The application of the constant multiple rule is demonstrated through various examples, including the differentiation of expressions like 4 sine x, 8 cosine x, and more complex combinations. The lesson also presents a rigorous proof of the derivative of sine x using the limit definition of a derivative and trigonometric identities, ultimately reinforcing the connection between calculus and trigonometry.

Takeaways
  • πŸ“š The derivative of sine (sin) x is cosine (cos) x.
  • πŸ“š The derivative of cosine (cos) x is negative sine (-sin) x.
  • πŸ“ˆ Using the constant multiple rule, the derivative of 4 sin x is 4 cos x.
  • πŸ“ˆ Similarly, the derivative of 8 cos x is -8 sin x.
  • πŸ”’ For the function 3 cos x - Ο€ sin x, the derivative is -3 sin x - Ο€ cos x by applying the constant multiple rule and sum/difference formulas for trigonometric functions.
  • πŸ“ When finding the derivative of sin x/5, rewrite the expression as (1/5) sin x and use the constant multiple rule to get (1/5) cos x.
  • πŸ’‘ To prove that the derivative of sin x (f'(x)) is cos x, use the limit definition of a derivative, f'(x) = lim(h->0) [f(x+h) - f(x)]/h.
  • πŸ“Š Apply sum and difference formulas for sine and cosine to simplify the expression for the derivative of sin x.
  • 🌟 Utilize the trigonometric limits: lim(x->0) (sin x)/x = 1 and lim(x->0) (1 - cos x)/x = 0 in the proof of the derivative of sin x.
  • πŸŽ“ Understanding the limit properties of trigonometric functions is crucial for differentiating trigonometric expressions.
  • πŸ“Œ The process of finding derivatives of trigonometric functions involves a combination of the constant multiple rule, sum/difference formulas, and limit properties.
Q & A

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

    -The derivative of the sine function with respect to x is the cosine function, denoted as d/dx(sine x) = cosine x.

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

    -The derivative of the cosine function with respect to x is the negative sine function, denoted as d/dx(cosine x) = -sine x.

  • How do you find the derivative of 4 times sine x?

    -Using the constant multiple rule, the derivative of 4 times sine x is 4 times the derivative of sine x, which is 4 times cosine x. So, the derivative is 4 cosine x.

  • How do you find the derivative of 8 times cosine x?

    -Using the constant multiple rule, the derivative of 8 times cosine x is 8 times the derivative of cosine x, which is 8 times negative sine x. Thus, the derivative is -8 sine x.

  • What is the derivative of 3 times cosine x minus pi times sine x?

    -The derivative of the expression 3 cosine x - pi sine x is found by applying the constant multiple rule to each term separately. The derivative of 3 cosine x is -3 sine x (since the derivative of cosine x is -sine x), and the derivative of pi sine x (with pi as a constant) is pi times the derivative of sine x, which is pi cosine x. Combining these, the derivative of the entire expression is -3 sine x - pi cosine x.

  • What is the derivative of sine x divided by 5?

    -Rewriting sine x divided by 5 as (1/5) times sine x, we apply the constant multiple rule. The derivative of (1/5) sine x is (1/5) times the derivative of sine x, which is (1/5) cosine x. So, the derivative is cosine x divided by 5.

  • How can we prove that the derivative of sine x is cosine x?

    -We can prove this by using the limit definition of a derivative. The derivative of a function f at x, denoted as f'(x), is the limit as h approaches 0 of [f(x + h) - f(x)] / h. For f(x) = sine x, we substitute and get [sine (x + h) - sine x] / h. Using the sum and difference formulas for sine, we simplify this expression and apply the special trigonometric limits to show that the limit as h approaches 0 of this expression is cosine x, thus proving that the derivative of sine x is indeed cosine x.

  • What are the two special trigonometric limits mentioned in the script?

    -The two special trigonometric limits mentioned are: the limit as x approaches 0 for sine x / x is equal to 1, and the limit as x approaches 0 for (1 - cosine x) / x is equal to 0.

  • How do the sum and difference formulas for sine and cosine help in differentiating trigonometric functions?

    -The sum and difference formulas for sine and cosine are essential in simplifying the expressions we get when differentiating trigonometric functions. They allow us to break down complex terms into simpler components, which can then be differentiated using the basic rules of differentiation. For example, when differentiating sine (x + h), we use the sum formula for sine to express it as sine x cosine h + cosine x sine h, which can then be differentiated term by term.

  • What is the constant multiple rule, and how is it applied in differentiation?

    -The constant multiple rule states that the derivative of a constant multiplied by a function is the constant multiplied by the derivative of that function. This rule is applied in differentiation by separating the constant from the variable function, finding the derivative of the function alone, and then multiplying the result by the constant. For example, the derivative of 4 sine x is found by first recognizing sine x's derivative as cosine x and then multiplying it by the constant 4, resulting in 4 cosine x.

  • Why is it important to know the limit definition of a derivative when working with trigonometric functions?

    -The limit definition of a derivative is fundamental because it provides the formal definition of a derivative and is particularly useful when dealing with more complex functions like trigonometric functions. It allows us to precisely define and calculate the rate of change at a specific point, which is crucial for understanding the behavior of the function. In the context of the script, using the limit definition helps in proving that the derivative of sine x is cosine x, which is a key concept in calculus and trigonometry.

Outlines
00:00
πŸ“š Derivatives of Sine and Cosine Functions

This paragraph introduces the concept of derivatives of sine and cosine functions. It explains that the derivative of sine x is cosine x and the derivative of cosine x is negative sine x. The paragraph then presents two problems to apply this knowledge: finding the derivative of 4 sine x and 8 cosine x. By using the constant multiple rule, the solutions are derived as 4 cosine x and negative 8 sine x, respectively. Additionally, the paragraph provides further examples to practice, such as finding the derivative of three cosine x minus pi sine x and differentiating sine x divided by five. The explanation emphasizes the importance of understanding the constant multiple rule and the sum and difference formulas for sine and cosine in solving these problems.

05:02
πŸ”’ Proof of Derivative of Sine as Cosine

This paragraph delves into the proof that the derivative of the function f(x) = sine x is indeed cosine x. It uses the limit definition of a derivative, which involves calculating the limit as h approaches zero of the expression (f(x+h) - f(x))/h. By substituting f(x) with sine x and applying the sum and difference formulas for sine, the expression is simplified. The paragraph then rearranges and separates the terms to handle the limit more effectively. Utilizing two key trigonometric limits, the limit of sine h/h as h approaches zero (which is one) and the limit of (1 - cosine x)/x as x approaches zero (which is zero), the proof concludes that the derivative of sine is cosine. This detailed explanation reinforces the understanding of the relationship between sine and cosine functions and their derivatives.

10:05
πŸ“‘ Additional Information

This paragraph, though brief, serves as a placeholder for additional content that may follow the detailed explanations and proofs provided in the previous paragraphs. It acknowledges the existence of more information to be covered, hinting at further exploration of the topic.

Mindmap
Keywords
πŸ’‘Derivatives
Derivatives are a fundamental concept in calculus that represent the rate of change or the slope of a function at a particular point. In the context of the video, the derivatives of sine and cosine functions are discussed, with the derivative of sine being cosine and the derivative of cosine being negative sine. This is crucial for understanding how these trigonometric functions behave and change as their inputs vary.
πŸ’‘Sine Function
The sine function is a trigonometric function that relates the angles of a right triangle to the ratios of the lengths of its sides. In the video, the sine function is used as a basis for discussing derivatives, with the key concept being that the derivative of sine with respect to x is the cosine function. This understanding is essential for analyzing the behavior of sine functions in various mathematical and real-world scenarios.
πŸ’‘Cosine Function
The cosine function, like the sine function, is a trigonometric function that plays a critical role in defining the ratios in right triangles and periodic phenomena. In the video, the derivative of the cosine function is explored, revealing that it is equal to negative sine. This property is vital for differentiating and analyzing the behavior of cosine functions within mathematical models.
πŸ’‘Constant Multiple Rule
The constant multiple rule is a basic differentiation rule stating that the derivative of a constant multiplied by a function is the constant multiplied by the derivative of that function. This rule is essential for simplifying and evaluating more complex derivatives in calculus. The video uses this rule to find the derivatives of expressions involving sine and cosine functions multiplied by constants.
πŸ’‘Sum and Difference Formulas
The sum and difference formulas for sine and cosine are trigonometric identities that describe how these functions change when added or subtracted. Specifically, for sine, the formula is sine(alpha + beta) = sine alpha cosine beta + cosine alpha sine beta. These formulas are crucial for simplifying trigonometric expressions and are used in the video to derive the proof for the derivative of the sine function.
πŸ’‘Limit Definition
The limit definition of a derivative is a formal way to define the derivative using limits. It states that the derivative of a function f at a point x is the limit as h approaches zero of the difference quotient [f(x + h) - f(x)]/h. This definition is foundational for understanding the concept of a derivative and is used in the video to prove that the derivative of sine x is cosine x.
πŸ’‘Trigonometric Limits
Trigonometric limits are special values that certain trigonometric expressions approach as their variables approach specific numbers. Two key limits mentioned in the video are the limit of sine x/x as x approaches zero (which is equal to 1) and the limit of (1 - cosine x)/x as x approaches zero (which is equal to 0). These limits are crucial for evaluating certain trigonometric expressions at specific points and are used in the proof of the derivative of sine x.
πŸ’‘Differentiation
Differentiation is the process of finding the derivative of a function, which describes how the function changes as its input variable changes. It is a core operation in calculus with numerous applications in physics, engineering, and other fields. The video focuses on differentiation of trigonometric functions, specifically sine and cosine, and provides rules and examples for finding their derivatives.
πŸ’‘Trigonometric Functions
Trigonometric functions, including sine and cosine, are mathematical functions that relate the angles of a right triangle to the ratios of its sides. They are widely used in mathematics, physics, and engineering to model periodic phenomena. The video delves into the derivatives of these functions, reinforcing their importance in understanding the behavior of trigonometric functions and their applications.
πŸ’‘Rate of Change
The rate of change is a concept in calculus that describes how the output of a function changes in response to changes in its input. The derivative of a function represents this rate of change at a specific point. In the context of the video, the derivatives of sine and cosine functions are used to analyze the rate at which these trigonometric functions change as their arguments vary.
Highlights

The lesson focuses on the derivatives of sine and cosine functions.

The derivative of sine x is cosine x.

The derivative of cosine x is negative sine x.

The derivative of 4 sine x is 4 cosine x using the constant multiple rule.

The derivative of 8 cosine x is negative 8 sine x.

The derivative of three cosine x minus pi sine x is negative 3 sine x minus pi cosine x.

The derivative of sine x divided by five is one-fifth cosine x or cosine x divided by 5.

To prove that the derivative of sine x is cosine x, the limit definition of a derivative is used.

The limit definition of a derivative involves the expression (f(x+h) - f(x))/h as h approaches 0.

Sine and cosine sum and difference formulas are essential for differentiating trigonometric functions.

The limit as h approaches 0 for sine h/h is equal to 1.

The limit as h approaches 0 for (1 - cosine x)/x is equal to 0.

These two special trigonometric limits are crucial for proving the derivative of sine is cosine.

The derivative of sine x is equal to cosine x, which is a fundamental result in calculus.

The lesson provides a clear and structured approach to understanding the derivatives of trigonometric functions.

Practical examples are used to illustrate the application of the constant multiple rule in differentiation.

The lesson demonstrates the process of differentiating composite trigonometric expressions.

Understanding the limit definition of a derivative is crucial for advanced calculus concepts.

Transcripts

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