Understanding the Problem and the Concept of Derivatives

Oct 13, 2025 by windailys.com 14 views

Understanding the Problem and the Concept of Derivatives

In this article, we will delve into the world of calculus and explore the concept of derivatives. Specifically, we will examine the given function $$y=\frac{2 x^3}{x2+1}$$ and find its derivative $$\frac{d y}{d x}$$. We will also explain why the function $$y$$ is an increasing function for $$x\ \textless \ 0$$.

The Given Function and Its Derivative

The given function is $$y=\frac{2 x^3}{x2+1}$$. To find its derivative, we will use the quotient rule of differentiation, which states that if $$y=\frac{u}{v}$$, then $$\frac{d y}{d x}=\frac{v \frac{d u}{d x}-u \frac{d v}{d x}}{v^2}$$.

Let's apply this rule to the given function. We have $$u=2 x^3$$ and $$v=x^2+1$$. To find $$\frac{d u}{d x}$$ and $$\frac{d v}{d x}$$, we will use the power rule of differentiation, which states that if $$y=x^n$$, then $$\frac{d y}{d x}=n x^{n-1}$$.

Applying the power rule, we get $$\frac{d u}{d x}=6 x^2$$ and $$\frac{d v}{d x}=2 x$$.

Now, we can substitute these values into the quotient rule formula:

\frac{d y}{d x}=\frac{(x^2+1)(6 x^2)-(2 x^3)(2 x)}{(x^2+1)^2}

Simplifying the expression, we get:

\frac{d y}{d x}=\frac{6 x^4+6 x^2-4 x^4}{(x^2+1)^2}

\frac{d y}{d x}=\frac{2 x^4+6 x^2}{(x^2+1)^2}

Why the Function is an Increasing Function

To determine why the function $$y$$ is an increasing function for $$x\ \textless \ 0$$, we need to examine the sign of the derivative $$\frac{d y}{d x}$$ in this interval.

Since $$x\ \textless \ 0$$, we know that $$x^2$$ is positive. Therefore, the denominator $$(x²⁺¹⁾2$$ is also positive.

Now, let's examine the numerator $$2 x^4+6 x^2$$. Since $$x\ \textless \ 0$$, we know that $$x^2$$ is positive. Therefore, the term $$2 x^4$$ is also positive.

The term $$6 x^2$$ is also positive since $$x^2$$ is positive.

Since both terms in the numerator are positive, the numerator is positive.

Therefore, the derivative $$\frac{d y}{d x}$$ is positive in the interval $$x\ \textless \ 0$$.

This means that the function $$y$$ is an increasing function in this interval.

Conclusion

In this article, we found the derivative of the given function $$y=\frac{2 x^3}{x2+1}$$ and explained why the function is an increasing function for $$x\ \textless \ 0$$. We used the quotient rule of differentiation and the power rule of differentiation to find the derivative, and we examined the sign of the derivative in the given interval to determine why the function is increasing.

Key Takeaways

The derivative of the function $$y=\frac{2 x^3}{x2+1}$$ is $$\frac{d y}{d x}=\frac{2 x^4+6 x^2}{(x2+1)^2}$$.
The function $$y$$ is an increasing function for $$x\ \textless \ 0$$.
The derivative $$\frac{d y}{d x}$$ is positive in the interval $$x\ \textless \ 0$$.

Further Reading

If you want to learn more about derivatives and how to find them, I recommend checking out the following resources:

Khan Academy: Derivatives
MIT OpenCourseWare: Calculus
Wolfram Alpha: Derivative Calculator

These resources provide a comprehensive introduction to derivatives and how to find them, as well as practice problems and examples to help you understand the concept better.
Q&A: Derivatives and Increasing Functions

In our previous article, we explored the concept of derivatives and how to find them. We also examined the given function $$y=\frac{2 x^3}{x2+1}$$ and found its derivative $$\frac{d y}{d x}$$. We also explained why the function $$y$$ is an increasing function for $$x\ \textless \ 0$$.

In this article, we will answer some frequently asked questions about derivatives and increasing functions.

Q: What is a derivative?

A: 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.

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

A: There are several rules of differentiation that you can use to find the derivative of a function. The most common rules are:

The power rule: If $$y=x^n$$, then $$\frac{d y}{d x}=n x^{n-1}$$.
The product rule: If $$y=u v$$, then $$\frac{d y}{d x}=u \frac{d v}{d x}+v \frac{d u}{d x}$$.
The quotient rule: If $$y=\frac{u}{v}$$, then $$\frac{d y}{d x}=\frac{v \frac{d u}{d x}-u \frac{d v}{d x}}{v^2}$$.

Q: What is an increasing function?

A: An increasing function is a function that gets larger as its input gets larger. In other words, if $$x_1 < x_2$$, then $$f(x_1) < f(x_2)$$.

Q: How do I determine if a function is increasing or decreasing?

A: To determine if a function is increasing or decreasing, you need to examine the sign of its derivative. If the derivative is positive, then the function is increasing. If the derivative is negative, then the function is decreasing.

Q: What is the significance of the derivative in real-world applications?

A: The derivative has many real-world applications, including:

Physics: The derivative is used to study the motion of objects and the forces that act upon them.
Engineering: The derivative is used to design and optimize systems, such as electrical circuits and mechanical systems.
Economics: The derivative is used to study the behavior of economic systems and to make predictions about future economic trends.

Q: Can you provide some examples of increasing and decreasing functions?

A: Yes, here are some examples:

Increasing function: $$f(x)=x^2$$
Decreasing function: $$f(x)=-x^2$$

Q: Can you provide some examples of functions that are neither increasing nor decreasing?

A: Yes, here are some examples:

Constant function: $$f(x)=c$$
Linear function: $$f(x)=mx+b$$

Conclusion

In this article, we answered some frequently asked questions about derivatives and increasing functions. We also provided some examples of increasing and decreasing functions, as well as functions that are neither increasing nor decreasing.

Key Takeaways

A derivative is a measure of how a function changes as its input changes.
The derivative is used to study the behavior of functions and to make predictions about future behavior.
An increasing function is a function that gets larger as its input gets larger.
The derivative is used to determine if a function is increasing or decreasing.
The derivative has many real-world applications, including physics, engineering, and economics.

Further Reading

If you want to learn more about derivatives and increasing functions, I recommend checking out the following resources:

Khan Academy: Derivatives
MIT OpenCourseWare: Calculus
Wolfram Alpha: Derivative Calculator

These resources provide a comprehensive introduction to derivatives and increasing functions, as well as practice problems and examples to help you understand the concept better.