Find $y^{\prime \prime}$.Given: $y = X(2x + 1)^5$\$y^{\prime \prime} = $[/tex\]

Introduction

In calculus, the second derivative of a function is a crucial concept that helps us understand the behavior of the function at a given point. It is a measure of how the rate of change of the function is changing. In this article, we will explore how to find the second derivative of a complex function, specifically the function $y = x(2x + 1)^5$.

The Function

The given function is $y = x(2x + 1)^5$. This function is a product of two functions: $x$ and $(2x + 1)^5$. To find the second derivative, we need to first find the first derivative.

Finding the First Derivative

To find the first derivative of the function, we can use the product rule of differentiation. The product rule states that if we have a function of the form $y = u(x)v(x)$, then the derivative of $y$ is given by $y^{\prime} = u^{\prime}(x)v(x) + u(x)v^{\prime}(x)$.

In this case, we have $y = x(2x + 1)^5$. Let $u(x) = x$ and $v(x) = (2x + 1)^5$. Then, we can find the derivatives of $u(x)$ and $v(x)$ as follows:

$$u^{\prime}(x) = \frac{d}{dx}x = 1$$

$$v^{\prime}(x) = \frac{d}{dx}(2x + 1)^5 = 5(2x + 1)^4 \cdot 2$$

Now, we can use the product rule to find the first derivative of $y$:

$$y^{\prime} = u^{\prime}(x)v(x) + u(x)v^{\prime}(x)$$

$$y^{\prime} = 1 \cdot (2x + 1)^5 + x \cdot 5(2x + 1)^4 \cdot 2$$

$$y^{\prime} = (2x + 1)^5 + 10x(2x + 1)^4$$

Finding the Second Derivative

To find the second derivative of $y$, we need to differentiate the first derivative $y^{\prime}$ with respect to $x$. We can use the product rule and the chain rule to find the second derivative.

Let's start by finding the derivative of $(2x + 1)^5$:

$$\frac{d}{dx}(2x + 1)^5 = 5(2x + 1)^4 \cdot 2$$

Now, we can find the derivative of $y^{\prime}$:

$$y^{\prime \prime} = \frac{d}{dx}((2x + 1)^5 + 10x(2x + 1)^4)$$

$$y^{\prime \prime} = 5(2x + 1)^4 \cdot 2 + 10(2x + 1)^4 + 10x \cdot 5(2x + 1)^4 \cdot 2$$

$$y^{\prime \prime} = 10(2x + 1)^4 + 10(2x + 1)^4 + 100x(2x + 1)^4$$

$$y^{\prime \prime} = 20(2x + 1)^4 + 100x(2x + 1)^4$$

$$y^{\prime \prime} = (2x + 1)^4(20 + 100x)$$

Conclusion

In this article, we have found the second derivative of the complex function $y = x(2x + 1)^5$. We used the product rule and the chain rule to find the first derivative, and then differentiated the first derivative to find the second derivative. The second derivative is given by $y^{\prime \prime} = (2x + 1)^4(20 + 100x)$.

Final Answer

Introduction

In our previous article, we explored how to find the second derivative of a complex function, specifically the function $y = x(2x + 1)^5$. We used the product rule and the chain rule to find the first derivative, and then differentiated the first derivative to find the second derivative. In this article, we will answer some common questions related to finding the second derivative of a complex function.

Q: What is the second derivative of a function?

A: The second derivative of a function is the derivative of the first derivative of the function. It is a measure of how the rate of change of the function is changing.

Q: How do I find the second derivative of a complex function?

A: To find the second derivative of a complex function, you need to first find the first derivative using the product rule and the chain rule. Then, you need to differentiate the first derivative to find the second derivative.

Q: What is the product rule of differentiation?

A: The product rule of differentiation states that if we have a function of the form $y = u(x)v(x)$, then the derivative of $y$ is given by $y^{\prime} = u^{\prime}(x)v(x) + u(x)v^{\prime}(x)$.

Q: What is the chain rule of differentiation?

A: The chain rule of differentiation states that if we have a function of the form $y = f(g(x))$, then the derivative of $y$ is given by $y^{\prime} = f^{\prime}(g(x)) \cdot g^{\prime}(x)$.

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

A: To apply the product rule and the chain rule, you need to identify the functions $u(x)$ and $v(x)$ in the given function. Then, you need to find the derivatives of $u(x)$ and $v(x)$ using the power rule and the chain rule. Finally, you need to substitute the derivatives into the product rule formula to find the first derivative, and then differentiate the first derivative to find the second derivative.

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

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

• Forgetting to apply the product rule and the chain rule
• Not identifying the functions $u(x)$ and $v(x)$ correctly
• Not finding the derivatives of $u(x)$ and $v(x)$ correctly
• Not substituting the derivatives into the product rule formula correctly
• Not differentiating the first derivative correctly to find the second derivative

Q: How can I practice finding the second derivative of a complex function?

A: You can practice finding the second derivative of a complex function by working through examples and exercises. You can also use online resources and calculators to check your work and get feedback.

Conclusion

In this article, we have answered some common questions related to finding the second derivative of a complex function. We have discussed the product rule and the chain rule, and provided tips and examples to help you practice finding the second derivative of a complex function. We hope this article has been helpful in your studies of calculus.

Final Answer

The final answer is $\boxed{(2x + 1)^4(20 + 100x)}$.