If Y = X 5 + 3 X 2 Y = X^5 + 3x^2 Y = X 5 + 3 X 2 , Find 2 D Y D X − X D 2 Y D X 2 \frac{2 \, D Y}{d X} - \frac{x \, D^2 Y}{d X^2} D X 2 D Y ​ − D X 2 X D 2 Y ​ .

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Introduction

In this article, we will explore the concept of finding the derivative of a function and then using it to evaluate a specific expression. The given function is $y = x^5 + 3x^2$, and we are asked to find the value of $\frac{2 \, d y}{d x} - \frac{x \, d^2 y}{d x^2}$. To solve this problem, we will first find the first and second derivatives of the given function.

Finding the First Derivative

The first derivative of a function $y = f(x)$ is denoted as $\frac{d y}{d x}$ and represents the rate of change of the function with respect to $x$. To find the first derivative of the given function, we will use the power rule of differentiation, which states that if $y = x^n$, then $\frac{d y}{d x} = n x^{n-1}$.

Using this rule, we can find the first derivative of the given function as follows:

$$\frac{d y}{d x} = \frac{d}{d x} (x^5 + 3x^2)$$

$$\frac{d y}{d x} = 5x^4 + 6x$$

Finding the Second Derivative

The second derivative of a function $y = f(x)$ is denoted as $\frac{d^2 y}{d x^2}$ and represents the rate of change of the first derivative with respect to $x$. To find the second derivative of the given function, we will differentiate the first derivative with respect to $x$.

Using the power rule of differentiation, we can find the second derivative of the given function as follows:

$$\frac{d^2 y}{d x^2} = \frac{d}{d x} (5x^4 + 6x)$$

$$\frac{d^2 y}{d x^2} = 20x^3 + 6$$

Evaluating the Expression

Now that we have found the first and second derivatives of the given function, we can evaluate the expression $\frac{2 \, d y}{d x} - \frac{x \, d^2 y}{d x^2}$.

Substituting the values of the first and second derivatives into the expression, we get:

$$\frac{2 \, d y}{d x} - \frac{x \, d^2 y}{d x^2} = 2(5x^4 + 6x) - x(20x^3 + 6)$$

Simplifying the expression, we get:

$$\frac{2 \, d y}{d x} - \frac{x \, d^2 y}{d x^2} = 10x^4 + 12x - 20x^4 - 6x$$

Combining like terms, we get:

$$\frac{2 \, d y}{d x} - \frac{x \, d^2 y}{d x^2} = -10x^4 + 6x$$

Conclusion

In this article, we have found the first and second derivatives of the given function $y = x^5 + 3x^2$. We have then used these derivatives to evaluate the expression $\frac{2 \, d y}{d x} - \frac{x \, d^2 y}{d x^2}$. The final answer is $-10x^4 + 6x$.

Final Answer

The final answer is $\boxed{-10x^4 + 6x}$.

References

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

Mathematical Operations

In this article, we have used the following mathematical operations:

• Power rule of differentiation
• Differentiation of a function
• Simplification of an expression

Key Concepts

• First derivative of a function
• Second derivative of a function
• Power rule of differentiation
• Differentiation of a function
• Simplification of an expression

Mathematical Formulas

• $\frac{d y}{d x} = n x^{n-1}$
• $\frac{d^2 y}{d x^2} = \frac{d}{d x} (\frac{d y}{d x})$
• $\frac{2 \, d y}{d x} - \frac{x \, d^2 y}{d x^2} = 2(5x^4 + 6x) - x(20x^3 + 6)$
  Q&A: If $y = x^5 + 3x^2$, find $\frac{2 \, d y}{d x} - \frac{x \, d^2 y}{d x^2}$ ===========================================================

Q: What is the first derivative of the function $y = x^5 + 3x^2$?

A: The first derivative of the function $y = x^5 + 3x^2$ is $\frac{d y}{d x} = 5x^4 + 6x$.

Q: What is the second derivative of the function $y = x^5 + 3x^2$?

A: The second derivative of the function $y = x^5 + 3x^2$ is $\frac{d^2 y}{d x^2} = 20x^3 + 6$.

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

A: To find the second derivative of a function, you need to differentiate the first derivative with respect to $x$. In this case, we differentiated the first derivative $5x^4 + 6x$ to get the second derivative $20x^3 + 6$.

Q: What is the expression $\frac{2 \, d y}{d x} - \frac{x \, d^2 y}{d x^2}$ equal to?

A: The expression $\frac{2 \, d y}{d x} - \frac{x \, d^2 y}{d x^2}$ is equal to $-10x^4 + 6x$.

Q: How do you simplify an expression like $\frac{2 \, d y}{d x} - \frac{x \, d^2 y}{d x^2}$?

A: To simplify an expression like $\frac{2 \, d y}{d x} - \frac{x \, d^2 y}{d x^2}$, you need to combine like terms and use the rules of algebra. In this case, we combined the like terms $10x^4 + 12x$ and $-20x^4 - 6x$ to get the simplified expression $-10x^4 + 6x$.

Q: What are the key concepts involved in finding the expression $\frac{2 \, d y}{d x} - \frac{x \, d^2 y}{d x^2}$?

A: The key concepts involved in finding the expression $\frac{2 \, d y}{d x} - \frac{x \, d^2 y}{d x^2}$ are:

• First derivative of a function
• Second derivative of a function
• Power rule of differentiation
• Differentiation of a function
• Simplification of an expression

Q: What are the mathematical operations involved in finding the expression $\frac{2 \, d y}{d x} - \frac{x \, d^2 y}{d x^2}$?

A: The mathematical operations involved in finding the expression $\frac{2 \, d y}{d x} - \frac{x \, d^2 y}{d x^2}$ are:

• Power rule of differentiation
• Differentiation of a function
• Simplification of an expression

Q: What are the mathematical formulas involved in finding the expression $\frac{2 \, d y}{d x} - \frac{x \, d^2 y}{d x^2}$?

A: The mathematical formulas involved in finding the expression $\frac{2 \, d y}{d x} - \frac{x \, d^2 y}{d x^2}$ are:

• $\frac{d y}{d x} = n x^{n-1}$
• $\frac{d^2 y}{d x^2} = \frac{d}{d x} (\frac{d y}{d x})$
• $\frac{2 \, d y}{d x} - \frac{x \, d^2 y}{d x^2} = 2(5x^4 + 6x) - x(20x^3 + 6)$

Q: What is the final answer to the problem?

A: The final answer to the problem is $\boxed{-10x^4 + 6x}$.

References

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

Mathematical Operations

In this article, we have used the following mathematical operations:

• Power rule of differentiation
• Differentiation of a function
• Simplification of an expression

Key Concepts

• First derivative of a function
• Second derivative of a function
• Power rule of differentiation
• Differentiation of a function
• Simplification of an expression

Mathematical Formulas

• $\frac{d y}{d x} = n x^{n-1}$
• $\frac{d^2 y}{d x^2} = \frac{d}{d x} (\frac{d y}{d x})$
• $\frac{2 \, d y}{d x} - \frac{x \, d^2 y}{d x^2} = 2(5x^4 + 6x) - x(20x^3 + 6)$