If Y = 3 X 2 + 1 Y=\sqrt{3x^2+1} Y = 3 X 2 + 1 ​ , Then D Y D X = \frac{dy}{dx}= D X D Y ​ = A. 1 2 6 X \frac{1}{2 \sqrt{6x}} 2 6 X ​ 1 ​ B. 1 2 3 X 2 + 1 \frac{1}{2 \sqrt{3x^2+1}} 2 3 X 2 + 1 ​ 1 ​ C. 3 X 3 X 2 + 1 \frac{3x}{\sqrt{3x^2+1}} 3 X 2 + 1 ​ 3 X ​ D. 3 X 3 X 2 + 1 3x \sqrt{3x^2+1} 3 X 3 X 2 + 1 ​

If $y=\sqrt{3x^2+1}$, then $\frac{dy}{dx}=$: A Comprehensive Analysis

In this article, we will delve into the world of calculus and explore the concept of differentiation. We will examine the given function $y=\sqrt{3x^2+1}$ and determine its derivative with respect to $x$. This will involve applying various rules and techniques of differentiation, including the chain rule and the power rule.

The given function is $y=\sqrt{3x^2+1}$. This can be rewritten as $y=(3x^2+1)^{\frac{1}{2}}$. We can see that the function is a composite function, consisting of a square root and a quadratic expression inside the square root.

To find the derivative of the function, we will apply the chain rule. The chain rule states that if we have a composite function of the form $y=f(g(x))$, then the derivative of $y$ with respect to $x$ is given by $\frac{dy}{dx}=\frac{df}{dg}\cdot\frac{dg}{dx}$.

In this case, we have $y=f(g(x))$, where $f(u)=(u)^{\frac{1}{2}}$ and $g(x)=3x^2+1$. We can find the derivatives of $f$ and $g$ with respect to their arguments:

$$\frac{df}{du}=\frac{1}{2}u^{-\frac{1}{2}}$$

$$\frac{dg}{dx}=6x$$

Now, we can apply the chain rule to find the derivative of $y$ with respect to $x$:

$$\frac{dy}{dx}=\frac{df}{dg}\cdot\frac{dg}{dx}=\frac{1}{2}(3x^2+1)^{-\frac{1}{2}}\cdot6x$$

We can simplify the derivative by combining the terms:

$$\frac{dy}{dx}=\frac{6x}{2(3x^2+1)^{\frac{1}{2}}}$$

$$\frac{dy}{dx}=\frac{3x}{(3x^2+1)^{\frac{1}{2}}}$$

In conclusion, we have found the derivative of the function $y=\sqrt{3x^2+1}$ with respect to $x$. The derivative is given by $\frac{dy}{dx}=\frac{3x}{(3x^2+1)^{\frac{1}{2}}}$. This is option C in the given multiple-choice question.

Let's compare our answer with the other options:

• Option A: $\frac{1}{2 \sqrt{6x}}$ - This is not the correct answer, as it does not match our result.
• Option B: $\frac{1}{2 \sqrt{3x^2+1}}$ - This is not the correct answer, as it does not match our result.
• Option D: $3x \sqrt{3x^2+1}$ - This is not the correct answer, as it does not match our result.

The final answer is option C: $\frac{3x}{\sqrt{3x^2+1}}$.
Q&A: If $y=\sqrt{3x^2+1}$, then $\frac{dy}{dx}=$: A Comprehensive Analysis

In our previous article, we explored the concept of differentiation and applied the chain rule to find the derivative of the function $y=\sqrt{3x^2+1}$. We determined that the derivative is given by $\frac{dy}{dx}=\frac{3x}{(3x^2+1)^{\frac{1}{2}}}$. In this article, we will address some common questions and concerns related to this topic.

A: The chain rule is a fundamental concept in calculus that allows us to find the derivative of a composite function. A composite function is a function of the form $y=f(g(x))$, where $f$ and $g$ are individual functions. The chain rule states that the derivative of $y$ with respect to $x$ is given by $\frac{dy}{dx}=\frac{df}{dg}\cdot\frac{dg}{dx}$. This means that we need to find the derivatives of $f$ and $g$ with respect to their arguments and then multiply them together.

A: To apply the chain rule, follow these steps:

1. Identify the composite function and the individual functions $f$ and $g$.
2. Find the derivatives of $f$ and $g$ with respect to their arguments.
3. Multiply the derivatives together to get the derivative of the composite function.

A: The power rule is a fundamental concept in calculus that allows us to find the derivative of a function of the form $y=x^n$. The power rule states that the derivative of $y$ with respect to $x$ is given by $\frac{dy}{dx}=nx^{n-1}$. This means that we need to multiply the exponent of $x$ by the coefficient of $x$ and then subtract 1 from the exponent.

A: To apply the power rule, follow these steps:

1. Identify the function and the exponent of $x$.
2. Multiply the exponent by the coefficient of $x$.
3. Subtract 1 from the exponent to get the new exponent.
4. Write the derivative as $nx^{n-1}$.

A: The chain rule and the power rule are two different concepts in calculus that allow us to find the derivative of a function. The chain rule is used to find the derivative of a composite function, while the power rule is used to find the derivative of a function of the form $y=x^n$. The chain rule involves multiplying the derivatives of the individual functions together, while the power rule involves multiplying the exponent by the coefficient of $x$ and then subtracting 1 from the exponent.

A: When finding the derivative of a function, you need to determine whether the function is a composite function or a function of the form $y=x^n$. If the function is a composite function, you should use the chain rule. If the function is of the form $y=x^n$, you should use the power rule.

In conclusion, the chain rule and the power rule are two fundamental concepts in calculus that allow us to find the derivative of a function. The chain rule is used to find the derivative of a composite function, while the power rule is used to find the derivative of a function of the form $y=x^n$. By understanding these concepts and how to apply them, you can find the derivative of a wide range of functions.