Find The Derivative Of X ( X 2 + 3 X − 2 ) 2 \sqrt{x}(x^2+3x-2)^2 X ​ ( X 2 + 3 X − 2 ) 2 .

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Introduction

In calculus, the derivative of a function is a measure of how the function changes as its input changes. 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 the given function $\sqrt{x}(x^2+3x-2)^2$. This involves applying the product rule and chain rule of differentiation, which are essential techniques in calculus.

Understanding the Function

The given function is $\sqrt{x}(x^2+3x-2)^2$. To find its derivative, we need to understand the structure of the function. It consists of two main components: $\sqrt{x}$ and $(x^2+3x-2)^2$. The first component is a square root function, while the second component is a quadratic function raised to the power of 2.

Applying the Product Rule

The product rule states that if we have a function of the form $f(x)g(x)$, then its derivative is given by $f'(x)g(x) + f(x)g'(x)$. In this case, we can apply the product rule to find the derivative of the given function.

Let $f(x) = \sqrt{x}$ and $g(x) = (x^2+3x-2)^2$. Then, we can find the derivatives of $f(x)$ and $g(x)$ separately.

Finding the Derivative of $f(x)$

To find the derivative of $f(x) = \sqrt{x}$, we can use the power rule of differentiation, which states that if $f(x) = x^n$, then $f'(x) = nx^{n-1}$. In this case, we have $f(x) = x^{1/2}$, so its derivative is $f'(x) = \frac{1}{2}x^{-1/2}$.

Finding the Derivative of $g(x)$

To find the derivative of $g(x) = (x^2+3x-2)^2$, we can use the chain rule of differentiation, which states that if we have a composite function of the form $f(g(x))$, then its derivative is given by $f'(g(x)) \cdot g'(x)$. In this case, we have $g(x) = (x^2+3x-2)^2$, so we need to find the derivative of the inner function $x^2+3x-2$ first.

Finding the Derivative of the Inner Function

To find the derivative of the inner function $x^2+3x-2$, we can use the power rule of differentiation. We have $f(x) = x^2+3x-2$, so its derivative is $f'(x) = 2x + 3$.

Applying the Chain Rule

Now that we have found the derivative of the inner function, we can apply the chain rule to find the derivative of $g(x) = (x^2+3x-2)^2$. We have $g(x) = (x^2+3x-2)^2$, so its derivative is given by $g'(x) = 2(x^2+3x-2)(2x+3)$.

Finding the Derivative of the Given Function

Now that we have found the derivatives of $f(x)$ and $g(x)$, we can apply the product rule to find the derivative of the given function $\sqrt{x}(x^2+3x-2)^2$. We have $f'(x) = \frac{1}{2}x^{-1/2}$ and $g'(x) = 2(x^2+3x-2)(2x+3)$, so the derivative of the given function is given by:

$$\frac{d}{dx}[\sqrt{x}(x^2+3x-2)^2] = \frac{1}{2}x^{-1/2}(x^2+3x-2)^2 + \sqrt{x} \cdot 2(x^2+3x-2)(2x+3)$$

Simplifying the Derivative

To simplify the derivative, we can expand the expression and combine like terms. We have:

$$\frac{d}{dx}[\sqrt{x}(x^2+3x-2)^2] = \frac{1}{2}x^{-1/2}(x^2+3x-2)^2 + 2\sqrt{x}(x^2+3x-2)(2x+3)$$

$$= \frac{1}{2}x^{-1/2}(x^2+3x-2)^2 + 2\sqrt{x}(2x^3+6x^2-4x+6x^2+9x-6)$$

$$= \frac{1}{2}x^{-1/2}(x^2+3x-2)^2 + 2\sqrt{x}(2x^3+12x^2+15x-6)$$

Conclusion

In this article, we have found the derivative of the given function $\sqrt{x}(x^2+3x-2)^2$ using the product rule and chain rule of differentiation. We have applied the power rule and chain rule to find the derivatives of the inner functions, and then used the product rule to find the derivative of the given function. The final derivative is given by:

$$\frac{d}{dx}[\sqrt{x}(x^2+3x-2)^2] = \frac{1}{2}x^{-1/2}(x^2+3x-2)^2 + 2\sqrt{x}(2x^3+12x^2+15x-6)$$

This derivative can be used to analyze the behavior of the given function and its applications in various fields.

References

• [1] Stewart, J. (2016). Calculus: Early Transcendentals. Cengage Learning.
• [2] Anton, H. (2018). Calculus: A New Horizon. John Wiley & Sons.
• [3] Edwards, C. H. (2019). Calculus: Early Transcendentals. Pearson Education.

Further Reading

• [1] Khan Academy. (n.d.). Calculus. Retrieved from https://www.khanacademy.org/math/calculus
• [2] MIT OpenCourseWare. (n.d.). Calculus. Retrieved from https://ocw.mit.edu/courses/mathematics/18-01-calculus-i-fall-2007/
• [3] Wolfram Alpha. (n.d.). Calculus. Retrieved from https://www.wolframalpha.com/calculators/calculus/
  

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Introduction

In our previous article, we found the derivative of the given function $\sqrt{x}(x^2+3x-2)^2$ using the product rule and chain rule of differentiation. In this article, we will address some common questions and concerns that readers may have regarding the derivative of this function.

Q: What is the product rule of differentiation?

A: The product rule of differentiation states that if we have a function of the form $f(x)g(x)$, then its derivative is given by $f'(x)g(x) + f(x)g'(x)$.

Q: What is the chain rule of differentiation?

A: The chain rule of differentiation states that if we have a composite function of the form $f(g(x))$, then its derivative is given by $f'(g(x)) \cdot g'(x)$.

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

A: To apply the product rule and chain rule, you need to identify the individual functions and their derivatives. Then, you can use the product rule to find the derivative of the product of the two functions, and the chain rule to find the derivative of the composite function.

Q: What is the derivative of $\sqrt{x}$?

A: The derivative of $\sqrt{x}$ is $\frac{1}{2}x^{-1/2}$.

Q: What is the derivative of $(x^2+3x-2)^2$?

A: The derivative of $(x^2+3x-2)^2$ is $2(x^2+3x-2)(2x+3)$.

Q: How do I simplify the derivative of $\sqrt{x}(x^2+3x-2)^2$?

A: To simplify the derivative, you can expand the expression and combine like terms.

Q: What is the final derivative of $\sqrt{x}(x^2+3x-2)^2$?

A: The final derivative of $\sqrt{x}(x^2+3x-2)^2$ is:

$$\frac{d}{dx}[\sqrt{x}(x^2+3x-2)^2] = \frac{1}{2}x^{-1/2}(x^2+3x-2)^2 + 2\sqrt{x}(2x^3+12x^2+15x-6)$$

Q: How can I use the derivative of $\sqrt{x}(x^2+3x-2)^2$ in real-world applications?

A: The derivative of $\sqrt{x}(x^2+3x-2)^2$ can be used to analyze the behavior of the given function and its applications in various fields, such as physics, engineering, and economics.

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

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

• Not identifying the individual functions and their derivatives
• Not applying the product rule and chain rule correctly
• Not simplifying the derivative
• Not checking the final answer for errors

Q: How can I practice finding the derivative of functions?

A: You can practice finding the derivative of functions by working through examples and exercises, such as those found in calculus textbooks or online resources.

Q: What are some resources for learning more about calculus and derivatives?

A: Some resources for learning more about calculus and derivatives include:

• Calculus textbooks, such as "Calculus: Early Transcendentals" by James Stewart
• Online resources, such as Khan Academy and MIT OpenCourseWare
• Calculus software, such as Wolfram Alpha

Conclusion

In this article, we have addressed some common questions and concerns that readers may have regarding the derivative of $\sqrt{x}(x^2+3x-2)^2$. We have provided explanations and examples to help readers understand the product rule and chain rule of differentiation, and how to apply them to find the derivative of a function. We hope that this article has been helpful in clarifying any confusion and providing a better understanding of the derivative of $\sqrt{x}(x^2+3x-2)^2$.