Find The Derivative Of The Function.${ \begin{array}{c} y = \sin 2\left(x 2 + 9\right) \ y^{\prime} = \square \end{array} }$

Introduction

In calculus, the derivative of a function represents the rate of change of the function with respect to its input. It is a fundamental concept in mathematics and has numerous applications in various fields, including physics, engineering, and economics. In this article, we will focus on finding the derivative of a trigonometric function, specifically the function $y = \sin^2\left(x^2 + 9\right)$. We will use the chain rule and other differentiation techniques to find the derivative of this function.

The Chain Rule

The chain rule is a fundamental rule in calculus that allows us to find the derivative of a composite function. A composite function is a function that is composed of two or more functions. The chain rule states that if we have a composite function of the form $f(g(x))$, then the derivative of this function is given by $f'(g(x)) \cdot g'(x)$.

Applying the Chain Rule to the Given Function

To find the derivative of the function $y = \sin^2\left(x^2 + 9\right)$, we can use the chain rule. We can rewrite the function as $y = (\sin(x^2 + 9))^2$. Now, we can apply the chain rule to find the derivative of this function.

Step 1: Find the Derivative of the Outer Function

The outer function is the square function, which is given by $f(u) = u^2$. The derivative of this function is given by $f'(u) = 2u$.

Step 2: Find the Derivative of the Inner Function

The inner function is the sine function, which is given by $g(x) = \sin(x^2 + 9)$. To find the derivative of this function, we can use the chain rule again. The derivative of the sine function is given by $g'(x) = \cos(x^2 + 9) \cdot (x^2 + 9)'$. Now, we need to find the derivative of the expression $(x^2 + 9)$.

Step 3: Find the Derivative of the Expression $(x^2 + 9)$

The derivative of the expression $(x^2 + 9)$ is given by $(x^2 + 9)' = 2x$.

Step 4: Substitute the Derivatives into the Chain Rule Formula

Now that we have found the derivatives of the outer and inner functions, we can substitute them into the chain rule formula. The derivative of the function $y = \sin^2\left(x^2 + 9\right)$ is given by:

$$y' = f'(g(x)) \cdot g'(x) = 2(\sin(x^2 + 9))^2 \cdot \cos(x^2 + 9) \cdot 2x$$

Simplifying the Derivative

We can simplify the derivative by combining the constants and the trigonometric functions. The derivative of the function $y = \sin^2\left(x^2 + 9\right)$ is given by:

$$y' = 4x \sin(x^2 + 9) \cos(x^2 + 9)$$

Conclusion

In this article, we have found the derivative of the function $y = \sin^2\left(x^2 + 9\right)$ using the chain rule and other differentiation techniques. We have broken down the problem into smaller steps and used the chain rule to find the derivative of the composite function. The final derivative is given by $y' = 4x \sin(x^2 + 9) \cos(x^2 + 9)$.

Example Problems

Here are some example problems that you can try to practice finding the derivative of a trigonometric function:

• Find the derivative of the function $y = \cos^2\left(x^2 + 9\right)$.
• Find the derivative of the function $y = \tan^2\left(x^2 + 9\right)$.
• Find the derivative of the function $y = \sin^2\left(x^3 + 9\right)$.

Tips and Tricks

Here are some tips and tricks that you can use to find the derivative of a trigonometric function:

• Use the chain rule to find the derivative of a composite function.
• Use the product rule to find the derivative of a product of two functions.
• Use the quotient rule to find the derivative of a quotient of two functions.
• Use the trigonometric identities to simplify the derivative.

Common Mistakes

Here are some common mistakes that you can avoid when finding the derivative of a trigonometric function:

• Not using the chain rule when finding the derivative of a composite function.
• Not using the product rule when finding the derivative of a product of two functions.
• Not using the quotient rule when finding the derivative of a quotient of two functions.
• Not simplifying the derivative using trigonometric identities.

Real-World Applications

The derivative of a trigonometric function has numerous real-world applications in various fields, including physics, engineering, and economics. Here are some examples of real-world applications:

• Finding the rate of change of a physical quantity, such as the velocity of an object.
• Finding the maximum or minimum value of a function, such as the maximum height of a projectile.
• Finding the equilibrium point of a system, such as the equilibrium point of a spring-mass system.

Conclusion

In conclusion, finding the derivative of a trigonometric function is an important concept in calculus that has numerous real-world applications. We have used the chain rule and other differentiation techniques to find the derivative of the function $y = \sin^2\left(x^2 + 9\right)$. We have also provided some example problems and tips and tricks to help you practice finding the derivative of a trigonometric function.

Introduction

In our previous article, we discussed how to find the derivative of a trigonometric function using the chain rule and other differentiation techniques. In this article, we will provide a Q&A section to help you better understand the concept of finding the derivative of a trigonometric function.

Q: What is the derivative of the function $y = \sin^2\left(x^2 + 9\right)$?

A: The derivative of the function $y = \sin^2\left(x^2 + 9\right)$ is given by $y' = 4x \sin(x^2 + 9) \cos(x^2 + 9)$.

Q: How do I find the derivative of a composite function?

A: To find the derivative of a composite function, you can use the chain rule. The chain rule states that if you have a composite function of the form $f(g(x))$, then the derivative of this function is given by $f'(g(x)) \cdot g'(x)$.

Q: What is the chain rule?

A: The chain rule is a fundamental rule in calculus that allows you to find the derivative of a composite function. It states that if you have a composite function of the form $f(g(x))$, then the derivative of this function is given by $f'(g(x)) \cdot g'(x)$.

Q: How do I find the derivative of a product of two functions?

A: To find the derivative of a product of two functions, you can use the product rule. The product rule states that if you have a product of two functions of the form $f(x) \cdot g(x)$, then the derivative of this function is given by $f'(x) \cdot g(x) + f(x) \cdot g'(x)$.

Q: How do I find the derivative of a quotient of two functions?

A: To find the derivative of a quotient of two functions, you can use the quotient rule. The quotient rule states that if you have a quotient of two functions of the form $\frac{f(x)}{g(x)}$, then the derivative of this function is given by $\frac{f'(x) \cdot g(x) - f(x) \cdot g'(x)}{g(x)^2}$.

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 using the chain rule when finding the derivative of a composite function.
• Not using the product rule when finding the derivative of a product of two functions.
• Not using the quotient rule when finding the derivative of a quotient of two functions.
• Not simplifying the derivative using trigonometric identities.

Q: What are some real-world applications of the derivative of a trigonometric function?

A: The derivative of a trigonometric function has numerous real-world applications in various fields, including physics, engineering, and economics. Some examples of real-world applications include:

• Finding the rate of change of a physical quantity, such as the velocity of an object.
• Finding the maximum or minimum value of a function, such as the maximum height of a projectile.
• Finding the equilibrium point of a system, such as the equilibrium point of a spring-mass system.

Q: How can I practice finding the derivative of a trigonometric function?

A: You can practice finding the derivative of a trigonometric function by working on example problems and exercises. Some tips for practicing include:

• Start with simple problems and gradually move on to more complex ones.
• Use online resources and calculators to check your work and get feedback.
• Practice finding the derivative of different types of trigonometric functions, such as sine, cosine, and tangent.

Q: What are some tips for simplifying the derivative of a trigonometric function?

A: Some tips for simplifying the derivative of a trigonometric function include:

• Use trigonometric identities to simplify the derivative.
• Combine like terms and simplify the expression.
• Use algebraic manipulations to simplify the derivative.

Conclusion

In conclusion, finding the derivative of a trigonometric function is an important concept in calculus that has numerous real-world applications. We have provided a Q&A section to help you better understand the concept of finding the derivative of a trigonometric function. We hope that this article has been helpful in answering your questions and providing you with a better understanding of the concept.