Find The Derivative Of X ( X 2 + 3 X − 2 ) 2 \sqrt{x}\left(x^2+3x-2\right)^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 a given function, $\sqrt{x}\left(x^2+3x-2\right)^2$. This function involves the product of two functions, $\sqrt{x}$ and $\left(x^2+3x-2\right)^2$, and we will use the product rule and chain rule to find its derivative.

The Product Rule

The product rule is a fundamental rule in calculus that states that if we have a function of the form $f(x)g(x)$, where $f(x)$ and $g(x)$ are both functions of $x$, then the derivative of $f(x)g(x)$ is given by:

$$\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$$

This rule can be extended to the product of more than two functions.

The Chain Rule

The chain rule is another fundamental rule in calculus that states that if we have a function of the form $f(g(x))$, where $f(x)$ and $g(x)$ are both functions of $x$, then the derivative of $f(g(x))$ is given by:

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

This rule can be extended to the composition of more than two functions.

Finding the Derivative of $\sqrt{x}\left(x^2+3x-2\right)^2$

To find the derivative of $\sqrt{x}\left(x^2+3x-2\right)^2$, we will use the product rule and chain rule. Let's first identify the two functions involved:

$$f(x) = \sqrt{x}$$

$$g(x) = \left(x^2+3x-2\right)^2$$

Now, we will find the derivatives of these two functions:

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

$$g'(x) = 2\left(x^2+3x-2\right) \cdot (2x+3)$$

Using the product rule, we can now find the derivative of $\sqrt{x}\left(x^2+3x-2\right)^2$:

$$\frac{d}{dx}[\sqrt{x}\left(x^2+3x-2\right)^2] = f'(x)g(x) + f(x)g'(x)$$

$$= \frac{1}{2\sqrt{x}}\left(x^2+3x-2\right)^2 + \sqrt{x} \cdot 2\left(x^2+3x-2\right) \cdot (2x+3)$$

Simplifying the expression, we get:

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

Simplifying the Expression

To simplify the expression, we can start by multiplying the numerator and denominator of the first term by $\sqrt{x}$:

$$\frac{\left(x^2+3x-2\right)^2}{2\sqrt{x}} = \frac{\left(x^2+3x-2\right)^2\sqrt{x}}{2x}$$

Now, we can combine the two terms:

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

Final Answer

The final answer is:

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

Conclusion

In this article, we have found the derivative of $\sqrt{x}\left(x^2+3x-2\right)^2$ using the product rule and chain rule. The final answer is a complex expression involving the product of two functions. We have also simplified the expression by multiplying the numerator and denominator of the first term by $\sqrt{x}$. This example illustrates the importance of using the product rule and chain rule in calculus to find the derivatives of complex functions.

References

• [1] Calculus, 3rd edition, Michael Spivak
• [2] Calculus, 2nd edition, James Stewart
• [3] Calculus, 1st edition, Michael Spivak

Further Reading

• [1] Derivative of a function
• [2] Product rule
• [3] Chain rule

Tags

• calculus
• derivative
• product rule
• chain rule
• mathematics
  

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Introduction

In our previous article, we found the derivative of $\sqrt{x}\left(x^2+3x-2\right)^2$ using the product rule and chain rule. However, we received many questions from readers who were struggling to understand the concept of derivatives and how to apply the product rule and chain rule. In this article, we will answer some of the most frequently asked questions about finding the derivative of $\sqrt{x}\left(x^2+3x-2\right)^2$.

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

A: The derivative of $\sqrt{x}$ is $\frac{1}{2\sqrt{x}}$. This is because the derivative of $x^n$ is $nx^{n-1}$, and in this case, $n = \frac{1}{2}$.

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

A: The derivative of $\left(x^2+3x-2\right)^2$ is $2\left(x^2+3x-2\right) \cdot (2x+3)$. This is because the derivative of $f(x)^n$ is $nf(x)^{n-1} \cdot f'(x)$, and in this case, $n = 2$.

Q: How do I apply the product rule to find the derivative of $\sqrt{x}\left(x^2+3x-2\right)^2$?

A: To apply the product rule, you need to find the derivatives of the two functions involved, $\sqrt{x}$ and $\left(x^2+3x-2\right)^2$. Then, you multiply the derivative of the first function by the second function, and the derivative of the second function by the first function. Finally, you add the two results together.

Q: How do I apply the chain rule to find the derivative of $\left(x^2+3x-2\right)^2$?

A: To apply the chain rule, you need to find the derivative of the outer function, which is $f(x) = x^2$, and the derivative of the inner function, which is $g(x) = x^2+3x-2$. Then, you multiply the derivative of the outer function by the derivative of the inner function.

Q: What is the final answer to the problem?

A: The final answer is:

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

Q: Can you provide more examples of how to apply the product rule and chain rule?

A: Yes, here are a few more examples:

• Find the derivative of $x^2 \cdot (3x+2)$ using the product rule.
• Find the derivative of $(x+1)^2 \cdot (2x-3)$ using the product rule and chain rule.
• Find the derivative of $\sqrt{x} \cdot (x^2-4)$ using the product rule and chain rule.

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:

• Forgetting to apply the product rule or chain rule when necessary.
• Making errors when simplifying the expression.
• Not checking the units of the answer.

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 in a textbook or online resource. You can also try to find the derivative of functions on your own, using the product rule and chain rule.

Conclusion

In this article, we have answered some of the most frequently asked questions about finding the derivative of $\sqrt{x}\left(x^2+3x-2\right)^2$. We have also provided additional examples and tips for practicing finding the derivative of functions. We hope that this article has been helpful in clarifying the concept of derivatives and how to apply the product rule and chain rule.

References

• [1] Calculus, 3rd edition, Michael Spivak
• [2] Calculus, 2nd edition, James Stewart
• [3] Calculus, 1st edition, Michael Spivak

Further Reading

• [1] Derivative of a function
• [2] Product rule
• [3] Chain rule

Tags

• calculus
• derivative
• product rule
• chain rule
• mathematics