Differentiate The Function.Given: F(x)=\ln \left(100 \sin ^2(x)\right ]Find: F ′ ( X ) = □ F^{\prime}(x)= \square F ′ ( X ) = □

Introduction

In this article, we will delve into the world of calculus and explore the concept of differentiation. We will focus on a specific function, $f(x)=\ln \left(100 \sin ^2(x)\right)$, and find its derivative, $f^{\prime}(x)$. Differentiation is a fundamental concept in mathematics, and it has numerous applications in various fields, including physics, engineering, and economics.

What is Differentiation?

Differentiation is a mathematical process that involves finding 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. In other words, it measures how fast the output of the function changes when one of its inputs changes.

The Chain Rule

To differentiate the given function, we will use the chain rule. The chain rule is a fundamental rule in calculus that allows us to differentiate composite functions. 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 the composite function is given by:

$$\frac{d}{dx}f(g(x)) = f^{\prime}(g(x)) \cdot g^{\prime}(x)$$

Differentiating the Given Function

Now, let's apply the chain rule to differentiate the given function, $f(x)=\ln \left(100 \sin ^2(x)\right)$. We can rewrite the function as:

$$f(x) = \ln \left(100 \sin ^2(x)\right) = \ln (100) + \ln \left(\sin ^2(x)\right)$$

Using the chain rule, we can differentiate the function as follows:

$$f^{\prime}(x) = \frac{d}{dx} \left(\ln (100) + \ln \left(\sin ^2(x)\right)\right)$$

$$f^{\prime}(x) = 0 + \frac{1}{\sin ^2(x)} \cdot 2 \sin (x) \cos (x)$$

$$f^{\prime}(x) = \frac{2 \sin (x) \cos (x)}{\sin ^2(x)}$$

Simplifying the Derivative

We can simplify the derivative by canceling out the common factor of $\sin (x)$ in the numerator and denominator:

$$f^{\prime}(x) = \frac{2 \cos (x)}{\sin (x)}$$

Conclusion

In this article, we differentiated the function $f(x)=\ln \left(100 \sin ^2(x)\right)$ using the chain rule. We found that the derivative of the function is $f^{\prime}(x) = \frac{2 \cos (x)}{\sin (x)}$. Differentiation is a powerful tool in mathematics, and it has numerous applications in various fields. We hope that this article has provided a comprehensive guide to differentiating the given function.

Final Answer

The final answer is $\boxed{\frac{2 \cos (x)}{\sin (x)}}$.

References

• [1] Calculus, 3rd edition, Michael Spivak
• [2] Calculus, 2nd edition, James Stewart
• [3] Differentiation, Khan Academy

Related Topics

• Differentiation of trigonometric functions
• Chain rule
• Derivatives of composite functions
• Calculus applications in physics and engineering

Frequently Asked Questions

• Q: What is differentiation? A: Differentiation is a mathematical process that involves finding the derivative of a function.
• Q: What is the chain rule? A: The chain rule is a fundamental rule in calculus that allows us to differentiate composite functions.
• Q: How do I differentiate a composite function? A: To differentiate a composite function, you can use the chain rule, which states that if we have a composite function of the form $f(g(x))$, then the derivative of the composite function is given by $f^{\prime}(g(x)) \cdot g^{\prime}(x)$.
  Differentiate the Function: A Comprehensive Guide =====================================================

Q&A: Differentiation and the Chain Rule

Q: What is differentiation?

A: Differentiation is a mathematical process that involves finding 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.

Q: What is the chain rule?

A: The chain rule is a fundamental rule in calculus that allows us to differentiate composite functions. 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 the composite function is given by:

$$\frac{d}{dx}f(g(x)) = f^{\prime}(g(x)) \cdot g^{\prime}(x)$$

Q: How do I differentiate a composite function?

A: To differentiate a composite function, you can use the chain rule. Here's a step-by-step guide:

1. Identify the outer function, $f(x)$, and the inner function, $g(x)$.
2. Find the derivative of the outer function, $f^{\prime}(x)$.
3. Find the derivative of the inner function, $g^{\prime}(x)$.
4. Substitute the inner function into the derivative of the outer function, $f^{\prime}(g(x))$.
5. Multiply the derivative of the outer function by the derivative of the inner function, $f^{\prime}(g(x)) \cdot g^{\prime}(x)$.

Q: What is the derivative of the function $f(x)=\ln \left(100 \sin ^2(x)\right)$?

A: To find the derivative of the function $f(x)=\ln \left(100 \sin ^2(x)\right)$, we can use the chain rule. Here's the step-by-step solution:

1. Identify the outer function, $f(x)=\ln (u)$, and the inner function, $u=100 \sin ^2(x)$.
2. Find the derivative of the outer function, $f^{\prime}(x)=\frac{1}{u}$.
3. Find the derivative of the inner function, $u^{\prime}(x)=200 \sin (x) \cos (x)$.
4. Substitute the inner function into the derivative of the outer function, $f^{\prime}(u)=\frac{1}{u}$.
5. Multiply the derivative of the outer function by the derivative of the inner function, $f^{\prime}(u) \cdot u^{\prime}(x)=\frac{1}{u} \cdot 200 \sin (x) \cos (x)$.
6. Simplify the expression, $f^{\prime}(x)=\frac{200 \sin (x) \cos (x)}{100 \sin ^2(x)}$.
7. Cancel out the common factor of $\sin (x)$ in the numerator and denominator, $f^{\prime}(x)=\frac{2 \cos (x)}{\sin (x)}$.

Q: What are some common mistakes to avoid when differentiating composite functions?

A: Here are some common mistakes to avoid when differentiating composite functions:

• Failing to identify the outer and inner functions.
• Failing to find the derivative of the outer function.
• Failing to find the derivative of the inner function.
• Failing to substitute the inner function into the derivative of the outer function.
• Failing to multiply the derivative of the outer function by the derivative of the inner function.

Q: How do I apply the chain rule to differentiate a function with multiple composite functions?

A: To apply the chain rule to differentiate a function with multiple composite functions, you can follow these steps:

1. Identify the outermost function, $f(x)$, and the innermost function, $g(x)$.
2. Find the derivative of the outermost function, $f^{\prime}(x)$.
3. Find the derivative of the innermost function, $g^{\prime}(x)$.
4. Substitute the innermost function into the derivative of the outermost function, $f^{\prime}(g(x))$.
5. Multiply the derivative of the outermost function by the derivative of the innermost function, $f^{\prime}(g(x)) \cdot g^{\prime}(x)$.
6. Repeat the process for each composite function, working from the innermost function to the outermost function.

Q: What are some real-world applications of the chain rule?

A: The chain rule has numerous real-world applications in fields such as physics, engineering, and economics. Here are a few examples:

• In physics, the chain rule is used to describe the motion of objects under the influence of forces.
• In engineering, the chain rule is used to design and optimize systems, such as electrical circuits and mechanical systems.
• In economics, the chain rule is used to model the behavior of economic systems, such as supply and demand curves.

Conclusion

In this article, we have explored the concept of differentiation and the chain rule. We have discussed how to differentiate composite functions using the chain rule and provided examples of common mistakes to avoid. We have also discussed real-world applications of the chain rule and provided a step-by-step guide to applying the chain rule to differentiate a function with multiple composite functions.