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 $\sqrt{x}(x^2 + 3x - 2)^2$.

Understanding the Function

The given function is a product of two functions: $\sqrt{x}$ and $(x^2 + 3x - 2)^2$. To find the derivative of this function, we will use the product rule of differentiation, which states that if $f(x) = u(x)v(x)$, then $f'(x) = u'(x)v(x) + u(x)v'(x)$.

Applying the Product Rule

To apply the product rule, we need to find the derivatives of the two functions $\sqrt{x}$ and $(x^2 + 3x - 2)^2$. The derivative of $\sqrt{x}$ is $\frac{1}{2\sqrt{x}}$, and the derivative of $(x^2 + 3x - 2)^2$ can be found using the chain rule.

Finding the Derivative of $(x^2 + 3x - 2)^2$

To find the derivative of $(x^2 + 3x - 2)^2$, we will use the chain rule, which states that if $f(x) = g(h(x))$, then $f'(x) = g'(h(x)) \cdot h'(x)$. In this case, $g(x) = x^2$ and $h(x) = (x^2 + 3x - 2)^2$.

Calculating the Derivative of $(x^2 + 3x - 2)^2$

Using the chain rule, we can find the derivative of $(x^2 + 3x - 2)^2$ as follows:

$$\frac{d}{dx}((x^2 + 3x - 2)^2) = 2(x^2 + 3x - 2) \cdot \frac{d}{dx}(x^2 + 3x - 2)$$

$$= 2(x^2 + 3x - 2) \cdot (2x + 3)$$

$$= 4x(x^2 + 3x - 2) + 6(x^2 + 3x - 2)$$

$$= 4x^3 + 12x^2 - 8x + 6x^2 + 18x - 12$$

$$= 4x^3 + 18x^2 + 10x - 12$$

Finding the Derivative of $\sqrt{x}$

The derivative of $\sqrt{x}$ is $\frac{1}{2\sqrt{x}}$.

Applying the Product Rule

Now that we have found the derivatives of the two functions, we can apply the product rule to find the derivative of the given function $\sqrt{x}(x^2 + 3x - 2)^2$.

$$\frac{d}{dx}(\sqrt{x}(x^2 + 3x - 2)^2) = \frac{1}{2\sqrt{x}}(x^2 + 3x - 2)^2 + \sqrt{x} \cdot 4x^3 + 18x^2 + 10x - 12$$

Simplifying the Derivative

To simplify the derivative, we can combine like terms and rewrite the expression in a more compact form.

$$\frac{d}{dx}(\sqrt{x}(x^2 + 3x - 2)^2) = \frac{(x^2 + 3x - 2)^2}{2\sqrt{x}} + \sqrt{x}(4x^3 + 18x^2 + 10x - 12)$$

Conclusion

In this article, we have found the derivative of the given function $\sqrt{x}(x^2 + 3x - 2)^2$ using the product rule of differentiation. We have also applied the chain rule to find the derivative of $(x^2 + 3x - 2)^2$. The final derivative is a combination of the derivatives of the two functions, and it can be simplified to a more compact form.

Final Answer

The final answer is:

$$\frac{d}{dx}(\sqrt{x}(x^2 + 3x - 2)^2) = \frac{(x^2 + 3x - 2)^2}{2\sqrt{x}} + \sqrt{x}(4x^3 + 18x^2 + 10x - 12)$$

Step-by-Step Solution

Here is the step-by-step solution to the problem:

1. Find the derivative of $\sqrt{x}$.
2. Find the derivative of $(x^2 + 3x - 2)^2$ using the chain rule.
3. Apply the product rule to find the derivative of $\sqrt{x}(x^2 + 3x - 2)^2$.
4. Simplify the derivative to a more compact form.

Key Concepts

• Product rule of differentiation
• Chain rule of differentiation
• Derivative of $\sqrt{x}$
• Derivative of $(x^2 + 3x - 2)^2$

Related Topics

• Calculus
• Differentiation
• Product rule
• Chain rule

References

• [1] Calculus by Michael Spivak
• [2] Differential Calculus by James Stewart
• [3] Calculus: Early Transcendentals by James Stewart
  

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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 of differentiation. In this article, we will answer some common questions related to finding the derivative of this function.

Q1: What is the product rule of differentiation?

A1: The product rule of differentiation is a rule in calculus that states that if $f(x) = u(x)v(x)$, then $f'(x) = u'(x)v(x) + u(x)v'(x)$. This rule is used to find the derivative of a product of two functions.

Q2: How do I apply the product rule to find the derivative of $\sqrt{x}(x^2 + 3x - 2)^2$?

A2: To apply the product rule, you need to find the derivatives of the two functions $\sqrt{x}$ and $(x^2 + 3x - 2)^2$. The derivative of $\sqrt{x}$ is $\frac{1}{2\sqrt{x}}$, and the derivative of $(x^2 + 3x - 2)^2$ can be found using the chain rule.

Q3: What is the chain rule of differentiation?

A3: The chain rule of differentiation is a rule in calculus that states that if $f(x) = g(h(x))$, then $f'(x) = g'(h(x)) \cdot h'(x)$. This rule is used to find the derivative of a composite function.

Q4: How do I find the derivative of $(x^2 + 3x - 2)^2$ using the chain rule?

A4: To find the derivative of $(x^2 + 3x - 2)^2$ using the chain rule, you need to find the derivatives of the outer function $x^2$ and the inner function $(x^2 + 3x - 2)^2$. The derivative of $x^2$ is $2x$, and the derivative of $(x^2 + 3x - 2)^2$ can be found using the product rule.

Q5: What is the final derivative of $\sqrt{x}(x^2 + 3x - 2)^2$?

A5: The final derivative of $\sqrt{x}(x^2 + 3x - 2)^2$ is $\frac{(x^2 + 3x - 2)^2}{2\sqrt{x}} + \sqrt{x}(4x^3 + 18x^2 + 10x - 12)$.

Q6: How do I simplify the derivative to a more compact form?

A6: To simplify the derivative, you can combine like terms and rewrite the expression in a more compact form.

Q7: What are some common mistakes to avoid when finding the derivative of a product of two functions?

A7: Some common mistakes to avoid when finding the derivative of a product of two functions include:

• Forgetting to apply the product rule
• Not finding the derivatives of both functions
• Not combining like terms when simplifying the derivative

Q8: How do I check my work when finding the derivative of a product of two functions?

A8: To check your work, you can:

• Plug in a test value into the derivative to see if it is correct
• Use a calculator to check the derivative
• Compare your work with the work of a classmate or tutor

Conclusion

In this article, we have answered some common questions related to finding the derivative of $\sqrt{x}(x^2 + 3x - 2)^2$. We have also provided some tips and tricks for finding the derivative of a product of two functions.

Final Answer

The final answer is:

$$\frac{d}{dx}(\sqrt{x}(x^2 + 3x - 2)^2) = \frac{(x^2 + 3x - 2)^2}{2\sqrt{x}} + \sqrt{x}(4x^3 + 18x^2 + 10x - 12)$$

Step-by-Step Solution

Here is the step-by-step solution to the problem:

1. Find the derivative of $\sqrt{x}$.
2. Find the derivative of $(x^2 + 3x - 2)^2$ using the chain rule.
3. Apply the product rule to find the derivative of $\sqrt{x}(x^2 + 3x - 2)^2$.
4. Simplify the derivative to a more compact form.

Key Concepts

• Product rule of differentiation
• Chain rule of differentiation
• Derivative of $\sqrt{x}$
• Derivative of $(x^2 + 3x - 2)^2$

Related Topics

• Calculus
• Differentiation
• Product rule
• Chain rule

References

• [1] Calculus by Michael Spivak
• [2] Differential Calculus by James Stewart
• [3] Calculus: Early Transcendentals by James Stewart