Find The Derivative Of The Function.$ F(x) = \sin^{-1}(4x) $ F'(x) = $

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Introduction

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In calculus, the derivative of a function represents the rate of change of the function with respect to its input. The derivative of the inverse sine function, denoted as $\sin^{-1}(x)$, is a fundamental concept in mathematics. In this article, we will explore the process of finding the derivative of the function $f(x) = \sin^{-1}(4x)$.

The Inverse Sine Function

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The inverse sine function, denoted as $\sin^{-1}(x)$, is the inverse of the sine function. It is defined as the angle whose sine is equal to a given value. The range of the inverse sine function is $[-\frac{\pi}{2}, \frac{\pi}{2}]$.

The Derivative of the Inverse Sine Function

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To find the derivative of the inverse sine function, we can use the chain rule and the fact that the derivative of the sine function is the cosine function. Let $f(x) = \sin^{-1}(x)$. Then, the derivative of $f(x)$ is given by:

$$f'(x) = \frac{1}{\sqrt{1-x^2}}$$

Finding the Derivative of $f(x) = \sin^{-1}(4x)$

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Now, let's find the derivative of the function $f(x) = \sin^{-1}(4x)$. We can use the chain rule to find the derivative of this function. 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:

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

In this case, we have $f(x) = \sin^{-1}(4x)$, so we can write:

$$f(x) = \sin^{-1}(4x)$$

$$f'(x) = \frac{1}{\sqrt{1-(4x)^2}} \cdot 4$$

Simplifying the Derivative

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Now, let's simplify the derivative of the function $f(x) = \sin^{-1}(4x)$. We can start by simplifying the expression inside the square root:

$$1-(4x)^2 = 1-16x^2$$

$$\sqrt{1-16x^2} = \sqrt{1-16x^2}$$

Now, we can substitute this expression back into the derivative:

$$f'(x) = \frac{4}{\sqrt{1-16x^2}}$$

Conclusion

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In this article, we have found the derivative of the function $f(x) = \sin^{-1}(4x)$. We used the chain rule and the fact that the derivative of the inverse sine function is the cosine function. The derivative of this function is given by:

$$f'(x) = \frac{4}{\sqrt{1-16x^2}}$$

This derivative can be used to find the rate of change of the function $f(x) = \sin^{-1}(4x)$ with respect to its input.

Example Problems

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Problem 1

Find the derivative of the function $f(x) = \sin^{-1}(3x)$.

Solution

We can use the chain rule to find the derivative of this function. 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:

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

In this case, we have $f(x) = \sin^{-1}(3x)$, so we can write:

$$f(x) = \sin^{-1}(3x)$$

$$f'(x) = \frac{1}{\sqrt{1-(3x)^2}} \cdot 3$$

Problem 2

Find the derivative of the function $f(x) = \sin^{-1}(2x)$.

Solution

We can use the chain rule to find the derivative of this function. 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:

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

In this case, we have $f(x) = \sin^{-1}(2x)$, so we can write:

$$f(x) = \sin^{-1}(2x)$$

$$f'(x) = \frac{1}{\sqrt{1-(2x)^2}} \cdot 2$$

Applications of the Derivative

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The derivative of the inverse sine function has many applications in mathematics and physics. Some of the applications include:

• Optimization: The derivative of the inverse sine function can be used to find the maximum or minimum of a function.
• Physics: The derivative of the inverse sine function can be used to describe the motion of an object in a circular path.
• Engineering: The derivative of the inverse sine function can be used to design and optimize systems that involve circular motion.

Conclusion

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In this article, we have found the derivative of the function $f(x) = \sin^{-1}(4x)$. We used the chain rule and the fact that the derivative of the inverse sine function is the cosine function. The derivative of this function is given by:

$$f'(x) = \frac{4}{\sqrt{1-16x^2}}$$

This derivative can be used to find the rate of change of the function $f(x) = \sin^{-1}(4x)$ with respect to its input. The derivative of the inverse sine function has many applications in mathematics and physics, and it is an important concept in calculus.

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Q: What is the derivative of the inverse sine function?

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A: The derivative of the inverse sine function is given by:

$$f'(x) = \frac{1}{\sqrt{1-x^2}}$$

Q: How do I find the derivative of the inverse sine function?

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A: To find the derivative of the inverse sine function, you can use the chain rule and the fact that the derivative of the sine function is the cosine function.

Q: What is the chain rule?

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A: The chain rule is a formula for finding the derivative of a composite function. It states that if we have a composite function of the form $f(g(x))$, then the derivative of this function is given by:

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

Q: How do I use the chain rule to find the derivative of the inverse sine function?

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A: To use the chain rule to find the derivative of the inverse sine function, you can follow these steps:

1. Identify the outer function and the inner function.
2. Find the derivative of the outer function.
3. Find the derivative of the inner function.
4. Multiply the derivatives of the outer and inner functions.

Q: What is the derivative of the function $f(x) = \sin^{-1}(4x)$?

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A: The derivative of the function $f(x) = \sin^{-1}(4x)$ is given by:

$$f'(x) = \frac{4}{\sqrt{1-16x^2}}$$

Q: How do I find the derivative of the function $f(x) = \sin^{-1}(4x)$?

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A: To find the derivative of the function $f(x) = \sin^{-1}(4x)$, you can use the chain rule and the fact that the derivative of the inverse sine function is the cosine function.

Q: What are some applications of the derivative of the inverse sine function?

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A: The derivative of the inverse sine function has many applications in mathematics and physics, including:

• Optimization: The derivative of the inverse sine function can be used to find the maximum or minimum of a function.
• Physics: The derivative of the inverse sine function can be used to describe the motion of an object in a circular path.
• Engineering: The derivative of the inverse sine function can be used to design and optimize systems that involve circular motion.

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

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A: To use the derivative of the inverse sine function in real-world applications, you can follow these steps:

1. Identify the problem you are trying to solve.
2. Determine the function that represents the problem.
3. Find the derivative of the function using the chain rule and the fact that the derivative of the inverse sine function is the cosine function.
4. Use the derivative to find the rate of change of the function with respect to its input.

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

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A: Some common mistakes to avoid when finding the derivative of the inverse sine function include:

• Forgetting to use the chain rule: The chain rule is a crucial step in finding the derivative of the inverse sine function.
• Not simplifying the expression: Simplifying the expression inside the square root can make it easier to find the derivative.
• Not checking the domain: The domain of the inverse sine function is $[-1, 1]$, so you need to check that the input is within this range.

Q: How do I check my work when finding the derivative of the inverse sine function?

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A: To check your work when finding the derivative of the inverse sine function, you can follow these steps:

1. Plug in a test value into the derivative.
2. Simplify the expression.
3. Check that the result is consistent with the original function.
4. Check that the result is consistent with the chain rule.

Q: What are some resources for learning more about the derivative of the inverse sine function?

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A: Some resources for learning more about the derivative of the inverse sine function include:

• Textbooks: There are many textbooks that cover the derivative of the inverse sine function, including "Calculus" by Michael Spivak and "Calculus: Early Transcendentals" by James Stewart.
• Online resources: There are many online resources that cover the derivative of the inverse sine function, including Khan Academy and MIT OpenCourseWare.
• Tutorials: There are many tutorials that cover the derivative of the inverse sine function, including video tutorials and interactive simulations.