Differentiate The Following With Respect To X X X : Y = X − X 2 − 2 X + 2 Y = X - \sqrt{x^2 - 2x + 2} Y = X − X 2 − 2 X + 2 ​

Introduction

In this article, we will be differentiating the given equation with respect to x. The equation is $y = x - \sqrt{x^2 - 2x + 2}$. We will use the chain rule and other differentiation techniques to find the derivative of this equation.

Understanding the Equation

The given equation is $y = x - \sqrt{x^2 - 2x + 2}$. This equation involves a square root term, which can be challenging to differentiate. To simplify the equation, we can start by expanding the square root term.

Expanding the Square Root Term

To expand the square root term, we can use the formula $\sqrt{a^2 + b^2} = a\sqrt{1 + \frac{b^2}{a^2}}$. In this case, we have $a = x - 1$ and $b = 1$. Therefore, we can write:

$$\sqrt{x^2 - 2x + 2} = \sqrt{(x - 1)^2 + 1}$$

Simplifying the Equation

Now that we have expanded the square root term, we can simplify the equation. We can rewrite the equation as:

$$y = x - \sqrt{(x - 1)^2 + 1}$$

Differentiating the Equation

To differentiate the equation, we can use the chain rule. 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 a composite function of the form $f(g(x)) = x - \sqrt{(x - 1)^2 + 1}$. We can identify the outer function as $f(u) = u - \sqrt{u^2 + 1}$ and the inner function as $g(x) = (x - 1)^2 + 1$.

Finding the Derivative of the Outer Function

To find the derivative of the outer function, we can use the chain rule. We have:

$$\frac{d}{du}f(u) = \frac{d}{du}(u - \sqrt{u^2 + 1}) = 1 - \frac{1}{2\sqrt{u^2 + 1}} \cdot 2u$$

Simplifying this expression, we get:

$$\frac{d}{du}f(u) = 1 - \frac{u}{\sqrt{u^2 + 1}}$$

Finding the Derivative of the Inner Function

To find the derivative of the inner function, we can use the power rule. We have:

$$\frac{d}{dx}g(x) = \frac{d}{dx}((x - 1)^2 + 1) = 2(x - 1)$$

Combining the Derivatives

Now that we have found the derivatives of the outer and inner functions, we can combine them using the chain rule. We have:

$$\frac{d}{dx}f(g(x)) = \frac{d}{du}f(u) \cdot \frac{d}{dx}g(x) = \left(1 - \frac{u}{\sqrt{u^2 + 1}}\right) \cdot 2(x - 1)$$

Substituting $u = (x - 1)^2 + 1$, we get:

$$\frac{d}{dx}f(g(x)) = \left(1 - \frac{(x - 1)^2 + 1}{\sqrt{((x - 1)^2 + 1)^2}}\right) \cdot 2(x - 1)$$

Simplifying this expression, we get:

$$\frac{d}{dx}f(g(x)) = \frac{2(x - 1)}{\sqrt{(x - 1)^2 + 2}}$$

Conclusion

In this article, we have differentiated the given equation with respect to x. We used the chain rule and other differentiation techniques to find the derivative of the equation. The final derivative is $\frac{2(x - 1)}{\sqrt{(x - 1)^2 + 2}}$.

Example Use Cases

The derivative of the given equation can be used in a variety of applications, such as:

• Finding the maximum or minimum value of the equation
• Determining the rate of change of the equation
• Solving optimization problems

Conclusion

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Frequently Asked Questions

In this article, we will be answering some of the most frequently asked questions about differentiating the given equation with respect to x.

Q: What is the given equation?

A: The given equation is $y = x - \sqrt{x^2 - 2x + 2}$.

Q: How do I differentiate the given equation with respect to x?

A: To differentiate the given equation with respect to x, we can use the chain rule and other differentiation techniques. We can start by expanding the square root term and then use the chain rule to find the derivative.

Q: What is the derivative of the given equation with respect to x?

A: The derivative of the given equation with respect to x is $\frac{2(x - 1)}{\sqrt{(x - 1)^2 + 2}}$.

Q: How do I use the derivative of the given equation in real-world applications?

A: The derivative of the given equation can be used in a variety of applications, such as finding the maximum or minimum value of the equation, determining the rate of change of the equation, and solving optimization problems.

Q: What are some common mistakes to avoid when differentiating the given equation with respect to x?

A: Some common mistakes to avoid when differentiating the given equation with respect to x include:

• Not expanding the square root term before differentiating
• Not using the chain rule correctly
• Not simplifying the derivative after finding it

Q: How do I simplify the derivative of the given equation?

A: To simplify the derivative of the given equation, we can start by factoring out any common terms and then simplifying the expression.

Q: What are some real-world applications of the derivative of the given equation?

A: Some real-world applications of the derivative of the given equation include:

• Finding the maximum or minimum value of a function
• Determining the rate of change of a function
• Solving optimization problems

Q: How do I find the maximum or minimum value of the given equation?

A: To find the maximum or minimum value of the given equation, we can use the derivative of the equation and set it equal to zero.

Q: What is the significance of the derivative of the given equation in calculus?

A: The derivative of the given equation is significant in calculus because it represents the rate of change of the function with respect to x. It can be used to find the maximum or minimum value of the function, determine the rate of change of the function, and solve optimization problems.

Conclusion

In conclusion, differentiating the given equation with respect to x is a complex process that requires the use of the chain rule and other differentiation techniques. The final derivative is $\frac{2(x - 1)}{\sqrt{(x - 1)^2 + 2}}$. This derivative can be used in a variety of applications, such as finding the maximum or minimum value of the equation, determining the rate of change of the equation, and solving optimization problems.