Find The Derivative Of The Function $y=\frac{3 \sin X}{2-9 \sin X}$. Y ′ ( X ) = Y^{\prime}(x)= Y ′ ( X ) =

Introduction

In calculus, the derivative of a function represents the rate of change of the function with respect to one of its variables. In this article, we will focus on finding the derivative of a given trigonometric function, which is $y=\frac{3 \sin x}{2-9 \sin x}.$ This function involves both sine and cosine functions, making it a bit more complex than the standard functions we encounter in calculus. We will use various techniques and formulas to simplify and find the derivative of this function.

The Quotient Rule

To find the derivative of the given function, we will use the quotient rule, which is a fundamental rule in calculus. The quotient rule states that if we have a function of the form $f(x)=\frac{g(x)}{h(x)},$ then the derivative of the function is given by $f^{{\prime}(x)=\frac{h(x)g}{\prime}(x)-g(x)h^{{\prime}(x)}{[h(x)]}{2}}.$ In our case, we have $g(x)=3 \sin x$ and $h(x)=2-9 \sin x.$ We will use these functions to find the derivative of the given function.

Finding the Derivative of the Function

Using the quotient rule, we can find the derivative of the function as follows:

$$y^{\prime}(x)=\frac{(2-9 \sin x)(3 \cos x)-(3 \sin x)(-9 \cos x)}{(2-9 \sin x)^{2}}.$$

Simplifying the Derivative

To simplify the derivative, we can expand and combine like terms:

$$y^{\prime}(x)=\frac{6 \cos x-27 \sin x \cos x+27 \sin x \cos x}{(2-9 \sin x)^{2}}.$$

Canceling Out Terms

We can cancel out the terms $27 \sin x \cos x$ in the numerator, which gives us:

$$y^{\prime}(x)=\frac{6 \cos x}{(2-9 \sin x)^{2}}.$$

Simplifying the Expression

We can simplify the expression by dividing both the numerator and the denominator by 2:

$$y^{\prime}(x)=\frac{3 \cos x}{(1-4.5 \sin x)^{2}}.$$

Conclusion

In this article, we found the derivative of the given trigonometric function using the quotient rule. We simplified the derivative by canceling out terms and dividing both the numerator and the denominator by 2. The final derivative is $y^{\prime}(x)=\frac{3 \cos x}{(1-4.5 \sin x)^{2}}.$ This derivative represents the rate of change of the function with respect to x.

Example Use Case

The derivative of the function can be used to find the rate of change of the function at a given point. For example, if we want to find the rate of change of the function at x=0, we can plug in x=0 into the derivative:

$$y^{\prime}(0)=\frac{3 \cos 0}{(1-4.5 \sin 0)^{2}}=\frac{3}{1^{2}}=3.$$

This means that the rate of change of the function at x=0 is 3.

Step-by-Step Solution

Here is a step-by-step solution to finding the derivative of the function:

1. Use the quotient rule to find the derivative of the function.
2. Simplify the derivative by canceling out terms and dividing both the numerator and the denominator by 2.
3. Use the simplified derivative to find the rate of change of the function at a given point.

Key Concepts

• The quotient rule is a fundamental rule in calculus that is used to find the derivative of a function.
• The derivative of a function represents the rate of change of the function with respect to one of its variables.
• The derivative of a trigonometric function can be found using various techniques and formulas.

Glossary

• Quotient Rule: A fundamental rule in calculus that is used to find the derivative of a function.
• Derivative: A measure of the rate of change of a function with respect to one of its variables.
• Trigonometric Function: A function that involves the sine, cosine, or tangent of an angle.

References

• [1] Calculus, 3rd edition, Michael Spivak.
• [2] Calculus, 2nd edition, James Stewart.

Conclusion

In this article, we found the derivative of a given trigonometric function using the quotient rule. We simplified the derivative by canceling out terms and dividing both the numerator and the denominator by 2. The final derivative is $y^{\prime}(x)=\frac{3 \cos x}{(1-4.5 \sin x)^{2}}.$ This derivative represents the rate of change of the function with respect to x.

Q: What is the quotient rule in calculus?

A: The quotient rule is a fundamental rule in calculus that is used to find the derivative of a function of the form $f(x)=\frac{g(x)}{h(x)}.$ The quotient rule states that the derivative of the function is given by $f^{{\prime}(x)=\frac{h(x)g}{\prime}(x)-g(x)h^{{\prime}(x)}{[h(x)]}{2}}.$

Q: How do I apply the quotient rule to find the derivative of a trigonometric function?

A: To apply the quotient rule, you need to identify the functions g(x) and h(x) in the given function. Then, you need to find the derivatives of g(x) and h(x) and plug them into the quotient rule formula. Finally, simplify the resulting expression to find the derivative of the function.

Q: What is the derivative of the function $y=\frac{3 \sin x}{2-9 \sin x}$?

A: The derivative of the function $y=\frac{3 \sin x}{2-9 \sin x}$ is given by $y^{\prime}(x)=\frac{3 \cos x}{(1-4.5 \sin x)^{2}}.$

Q: How do I simplify the derivative of a trigonometric function?

A: To simplify the derivative of a trigonometric function, you can cancel out terms, combine like terms, and divide both the numerator and the denominator by a common factor.

Q: What is the rate of change of the function $y=\frac{3 \sin x}{2-9 \sin x}$ at x=0?

A: The rate of change of the function $y=\frac{3 \sin x}{2-9 \sin x}$ at x=0 is given by $y^{\prime}(0)=\frac{3 \cos 0}{(1-4.5 \sin 0)^{{2}}=\frac{3}{1}{2}}=3.$

Q: What are some common mistakes to avoid when finding the derivative of a trigonometric function?

A: Some common mistakes to avoid when finding the derivative of a trigonometric function include:

• Not identifying the functions g(x) and h(x) correctly
• Not finding the derivatives of g(x) and h(x) correctly
• Not simplifying the resulting expression correctly
• Not canceling out terms correctly

Q: How do I use the derivative of a trigonometric function in real-world applications?

A: The derivative of a trigonometric function can be used to find the rate of change of the function at a given point. This can be useful in a variety of real-world applications, such as:

• Finding the rate of change of a population over time
• Finding the rate of change of a physical quantity, such as velocity or acceleration
• Finding the rate of change of a financial quantity, such as stock prices or interest rates

Q: What are some common trigonometric functions that are used in calculus?

A: Some common trigonometric functions that are used in calculus include:

• Sine (sin x)
• Cosine (cos x)
• Tangent (tan x)
• Secant (sec x)
• Cosecant (csc x)
• Cotangent (cot x)

Q: How do I find the derivative of a trigonometric function using a calculator?

A: To find the derivative of a trigonometric function using a calculator, you can use the following steps:

1. Enter the function into the calculator.
2. Use the derivative function on the calculator to find the derivative of the function.
3. Simplify the resulting expression to find the derivative of the function.

Note: The specific steps may vary depending on the calculator being used.

Q: What are some common applications of trigonometric functions in calculus?

A: Some common applications of trigonometric functions in calculus include:

• Finding the rate of change of a function
• Finding the maximum and minimum values of a function
• Finding the area under a curve
• Finding the volume of a solid
• Finding the surface area of a solid

Q: How do I use trigonometric functions to model real-world phenomena?

A: Trigonometric functions can be used to model a variety of real-world phenomena, including:

• The motion of a pendulum
• The motion of a spring
• The motion of a wave
• The motion of a particle
• The growth of a population

Q: What are some common challenges when working with trigonometric functions in calculus?

A: Some common challenges when working with trigonometric functions in calculus include:

• Simplifying complex expressions
• Finding the derivatives of trigonometric functions
• Using trigonometric functions to model real-world phenomena
• Interpreting the results of trigonometric functions in real-world contexts

Q: How do I overcome common challenges when working with trigonometric functions in calculus?

A: To overcome common challenges when working with trigonometric functions in calculus, you can:

• Practice simplifying complex expressions
• Practice finding the derivatives of trigonometric functions
• Practice using trigonometric functions to model real-world phenomena
• Practice interpreting the results of trigonometric functions in real-world contexts

Q: What are some common resources for learning about trigonometric functions in calculus?

A: Some common resources for learning about trigonometric functions in calculus include:

• Textbooks on calculus
• Online resources, such as Khan Academy and MIT OpenCourseWare
• Calculus courses at a local college or university
• Calculus software, such as Mathematica and Maple

Q: How do I choose the right resources for learning about trigonometric functions in calculus?

A: To choose the right resources for learning about trigonometric functions in calculus, you can:

• Consider your learning style and preferences
• Consider the level of difficulty of the resource
• Consider the relevance of the resource to your goals and interests
• Consider the cost and accessibility of the resource

Q: What are some common mistakes to avoid when learning about trigonometric functions in calculus?

A: Some common mistakes to avoid when learning about trigonometric functions in calculus include:

• Not practicing enough
• Not reviewing material regularly
• Not seeking help when needed
• Not using a variety of resources
• Not taking breaks and staying motivated

Q: How do I stay motivated when learning about trigonometric functions in calculus?

A: To stay motivated when learning about trigonometric functions in calculus, you can:

• Set achievable goals and rewards
• Find a study group or tutor
• Use a variety of resources and learning tools
• Take breaks and engage in activities you enjoy
• Celebrate your progress and accomplishments