Differentiate: Y = Ln ⁡ X X 4 Y = \frac{\ln X}{x^4} Y = X 4 L N X ​

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Introduction

In calculus, differentiation is a fundamental concept that deals with finding the rate of change of a function with respect to one of its variables. In this article, we will focus on differentiating the function $y = \frac{\ln x}{x^4}$, which involves applying various differentiation rules and techniques.

The Function

The given function is $y = \frac{\ln x}{x^4}$. This function involves a natural logarithm in the numerator and a power function in the denominator. To differentiate this function, we need to apply the quotient rule and the chain rule of differentiation.

Quotient Rule

The quotient rule states that if we have a function of the form $y = \frac{u}{v}$, where $u$ and $v$ are both functions of $x$, then the derivative of $y$ with respect to $x$ is given by:

$$\frac{dy}{dx} = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}$$

In our case, $u = \ln x$ and $v = x^4$. We need to find the derivatives of $u$ and $v$ with respect to $x$.

Derivative of $u = \ln x$

The derivative of $u = \ln x$ with respect to $x$ is given by:

$$\frac{du}{dx} = \frac{1}{x}$$

Derivative of $v = x^4$

The derivative of $v = x^4$ with respect to $x$ is given by:

$$\frac{dv}{dx} = 4x^3$$

Applying the Quotient Rule

Now that we have the derivatives of $u$ and $v$, we can apply the quotient rule to find the derivative of $y = \frac{\ln x}{x^4}$:

$$\frac{dy}{dx} = \frac{x^4\frac{1}{x} - \ln x(4x^3)}{(x^4)^2}$$

Simplifying the expression, we get:

$$\frac{dy}{dx} = \frac{x^3 - 4x^3\ln x}{x^8}$$

Simplifying the Expression

We can simplify the expression further by factoring out a common term:

$$\frac{dy}{dx} = \frac{x^3(1 - 4\ln x)}{x^8}$$

$$\frac{dy}{dx} = \frac{1 - 4\ln x}{x^5}$$

Conclusion

In this article, we differentiated the function $y = \frac{\ln x}{x^4}$ using the quotient rule and the chain rule of differentiation. We found that the derivative of the function is $\frac{1 - 4\ln x}{x^5}$. This result can be used to find the rate of change of the function with respect to $x$.

Example Problems

1. Find the derivative of the function $y = \frac{\ln x}{x^2}$.
2. Find the derivative of the function $y = \frac{\ln x}{x^6}$.
3. Find the derivative of the function $y = \frac{x^2}{\ln x}$.

Solution to Example Problems

1. Using the quotient rule, we can find the derivative of the function $y = \frac{\ln x}{x^2}$:

$$\frac{dy}{dx} = \frac{x^2\frac{1}{x} - \ln x(2x)}{(x^2)^2}$$

Simplifying the expression, we get:

$$\frac{dy}{dx} = \frac{x - 2x\ln x}{x^4}$$

2. Using the quotient rule, we can find the derivative of the function $y = \frac{\ln x}{x^6}$:

$$\frac{dy}{dx} = \frac{x^6\frac{1}{x} - \ln x(6x^5)}{(x^6)^2}$$

Simplifying the expression, we get:

$$\frac{dy}{dx} = \frac{x^5 - 6x^5\ln x}{x^{12}}$$

3. Using the quotient rule, we can find the derivative of the function $y = \frac{x^2}{\ln x}$:

$$\frac{dy}{dx} = \frac{\ln x(2x) - x^2\frac{1}{x}}{(\ln x)^2}$$

Simplifying the expression, we get:

$$\frac{dy}{dx} = \frac{2x\ln x - x}{(\ln x)^2}$$

Final Answer

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Frequently Asked Questions

In this article, we will answer some frequently asked questions related to differentiating the function $y = \frac{\ln x}{x^4}$.

Q: What is the derivative of $y = \frac{\ln x}{x^4}$?

A: The derivative of $y = \frac{\ln x}{x^4}$ is $\frac{1 - 4\ln x}{x^5}$.

Q: How do I apply the quotient rule to differentiate $y = \frac{\ln x}{x^4}$?

A: To apply the quotient rule, you need to find the derivatives of the numerator and denominator separately. The derivative of the numerator is $\frac{1}{x}$ and the derivative of the denominator is $4x^3$. Then, you can use the quotient rule formula to find the derivative of the function.

Q: What is the quotient rule formula?

A: The quotient rule formula is:

$$\frac{dy}{dx} = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}$$

Q: How do I simplify the expression after applying the quotient rule?

A: After applying the quotient rule, you can simplify the expression by factoring out common terms and canceling out any common factors.

Q: Can I use the chain rule to differentiate $y = \frac{\ln x}{x^4}$?

A: Yes, you can use the chain rule to differentiate $y = \frac{\ln x}{x^4}$. However, in this case, it is more efficient to use the quotient rule.

Q: What is the significance of the derivative of $y = \frac{\ln x}{x^4}$?

A: The derivative of $y = \frac{\ln x}{x^4}$ represents the rate of change of the function with respect to $x$. It can be used to find the maximum and minimum values of the function, as well as the points of inflection.

Q: How do I find the maximum and minimum values of $y = \frac{\ln x}{x^4}$?

A: To find the maximum and minimum values of $y = \frac{\ln x}{x^4}$, you need to find the critical points of the function by setting the derivative equal to zero and solving for $x$. Then, you can use the second derivative test to determine whether the critical points correspond to maximum or minimum values.

Q: What is the second derivative test?

A: The second derivative test is a method used to determine whether a critical point corresponds to a maximum or minimum value. If the second derivative is positive at a critical point, then the point corresponds to a minimum value. If the second derivative is negative at a critical point, then the point corresponds to a maximum value.

Q: Can I use the second derivative test to find the points of inflection of $y = \frac{\ln x}{x^4}$?

A: Yes, you can use the second derivative test to find the points of inflection of $y = \frac{\ln x}{x^4}$. A point of inflection is a point where the concavity of the function changes. To find the points of inflection, you need to find the values of $x$ where the second derivative changes sign.

Conclusion

In this article, we answered some frequently asked questions related to differentiating the function $y = \frac{\ln x}{x^4}$. We also discussed the significance of the derivative and how to use it to find the maximum and minimum values of the function, as well as the points of inflection.