Let F ( X ) = 2 X 2 + 2 F(x)=\sqrt{2x^2+2} F ( X ) = 2 X 2 + 2 ​ Find F ′ ( X F^{\prime}(x F ′ ( X ].

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Introduction

In this article, we will explore the concept of finding the derivative of a square root function. The square root function is a fundamental concept in mathematics, and its derivative is a crucial tool in calculus. We will use the given function $f(x)=\sqrt{2x^2+2}$ to demonstrate the process of finding the derivative of a square root function.

What is a Derivative?

A derivative is a measure of how a function changes as its input changes. It is a fundamental concept in calculus and is used to study the behavior of functions. The derivative of a function $f(x)$ is denoted as $f^{\prime}(x)$ and is defined as:

$$f^{\prime}(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$

Finding the Derivative of a Square Root Function

To find the derivative of a square root function, we can use the chain rule and the power rule. 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:

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

The power rule states that if we have a function of the form $f(x) = x^n$, then the derivative of the function is given by:

$$\frac{d}{dx} x^n = n \cdot x^{n-1}$$

Applying the Chain Rule and Power Rule

To find the derivative of the given function $f(x)=\sqrt{2x^2+2}$, we can use the chain rule and the power rule. We can rewrite the function as:

$$f(x) = (2x^2+2)^{\frac{1}{2}}$$

Using the chain rule, we can write:

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

Using the power rule, we can simplify the expression:

$$f^{\prime}(x) = \frac{2x}{\sqrt{2x^2+2}}$$

Simplifying the Derivative

We can simplify the derivative by rationalizing the denominator. To rationalize the denominator, we can multiply the numerator and denominator by the conjugate of the denominator:

$$f^{\prime}(x) = \frac{2x}{\sqrt{2x^2+2}} \cdot \frac{\sqrt{2x^2+2}}{\sqrt{2x^2+2}}$$

Simplifying the expression, we get:

$$f^{\prime}(x) = \frac{2x\sqrt{2x^2+2}}{2x^2+2}$$

Conclusion

In this article, we have explored the concept of finding the derivative of a square root function. We have used the given function $f(x)=\sqrt{2x^2+2}$ to demonstrate the process of finding the derivative of a square root function. We have applied the chain rule and the power rule to find the derivative and simplified the expression by rationalizing the denominator.

Final Answer

The final answer is $\boxed{\frac{2x\sqrt{2x^2+2}}{2x^2+2}}$.

References

• [1] Calculus by Michael Spivak
• [2] Calculus by James Stewart
• [3] Derivatives by Khan Academy

Related Topics

• Finding the derivative of a polynomial function
• Finding the derivative of a trigonometric function
• Finding the derivative of an exponential function

Frequently Asked Questions

• Q: What is the derivative of a square root function? A: The derivative of a square root function is given by the formula $f^{\prime}(x) = \frac{1}{2} (2x^2+2)^{-\frac{1}{2}} \cdot 4x$.
• Q: How do I simplify the derivative of a square root function? A: To simplify the derivative of a square root function, you can rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator.
  Derivative of a Square Root Function: Q&A =============================================

Introduction

In our previous article, we explored the concept of finding the derivative of a square root function. We used the given function $f(x)=\sqrt{2x^2+2}$ to demonstrate the process of finding the derivative of a square root function. In this article, we will answer some frequently asked questions about the derivative of a square root function.

Q&A

Q: What is the derivative of a square root function?

A: The derivative of a square root function is given by the formula $f^{\prime}(x) = \frac{1}{2} (2x^2+2)^{-\frac{1}{2}} \cdot 4x$.

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

A: To simplify the derivative of a square root function, you can rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator.

Q: What is the final answer for the derivative of the given function $f(x)=\sqrt{2x^2+2}$?

A: The final answer is $\boxed{\frac{2x\sqrt{2x^2+2}}{2x^2+2}}$.

Q: Can I use the power rule to find the derivative of a square root function?

A: No, you cannot use the power rule to find the derivative of a square root function. The power rule is used to find the derivative of a function of the form $f(x) = x^n$, but a square root function is not in this form.

Q: Can I use the chain rule to find the derivative of a square root function?

A: Yes, you can use the chain rule to find the derivative of a square root function. 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 $f^{\prime}(g(x)) \cdot g^{\prime}(x)$.

Q: How do I apply the chain rule to find the derivative of a square root function?

A: To apply the chain rule to find the derivative of a square root function, you need to identify the inner and outer functions. The inner function is $2x^2+2$ and the outer function is the square root function. Then, you can use the chain rule to find the derivative of the composite function.

Q: What is the significance of the derivative of a square root function?

A: The derivative of a square root function is significant because it helps us understand how the function changes as its input changes. It is a fundamental concept in calculus and is used to study the behavior of functions.

Conclusion

In this article, we have answered some frequently asked questions about the derivative of a square root function. We have provided explanations and examples to help you understand the concept of finding the derivative of a square root function.

Final Answer

The final answer is $\boxed{\frac{2x\sqrt{2x^2+2}}{2x^2+2}}$.

References

• [1] Calculus by Michael Spivak
• [2] Calculus by James Stewart
• [3] Derivatives by Khan Academy

Related Topics

• Finding the derivative of a polynomial function
• Finding the derivative of a trigonometric function
• Finding the derivative of an exponential function

Frequently Asked Questions

• Q: What is the derivative of a square root function? A: The derivative of a square root function is given by the formula $f^{\prime}(x) = \frac{1}{2} (2x^2+2)^{-\frac{1}{2}} \cdot 4x$.
• Q: How do I simplify the derivative of a square root function? A: To simplify the derivative of a square root function, you can rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator.