Finding the Derivative of the Logarithmic Term

Oct 13, 2025 by magicalceylontours.com 104 views

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. It is a crucial tool in various fields, including physics, engineering, and economics. In this article, we will focus on differentiating the function $y = \frac{\ln x}{x^6}$ , which is a rational function involving a logarithmic term.

What is Differentiation?

Differentiation is a mathematical operation that finds the derivative of a function. The derivative of a function represents the rate of change of the function with respect to one of its variables. It is denoted by the symbol $\frac{dy}{dx}$ or $y'$ . The derivative of a function can be thought of as the slope of the tangent line to the graph of the function at a given point.

The Chain Rule

To differentiate the function $y = \frac{\ln x}{x^6}$ , we will use the chain rule, which is a fundamental rule in calculus. 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} = f'(g(x)) \cdot g'(x)

Applying the Chain Rule

To differentiate the function $y = \frac{\ln x}{x^6}$ , we can rewrite it as:

y = \frac{\ln x}{x^6} = \frac{1}{x^6} \cdot \ln x

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

\frac{dy}{dx} = \frac{d}{dx} \left( \frac{1}{x^6} \right) \cdot \ln x + \frac{1}{x^6} \cdot \frac{d}{dx} (\ln x)

Finding the Derivative of the Logarithmic Term

To find the derivative of the logarithmic term $\ln x$ , we can use the fact that the derivative of $\ln x$ is $\frac{1}{x}$ . Therefore, we have:

\frac{d}{dx} (\ln x) = \frac{1}{x}

Finding the Derivative of the Rational Term

To find the derivative of the rational term $\frac{1}{x^6}$ , we can use the power rule of differentiation, which states that if we have a function of the form $f(x) = x^n$ , then the derivative of $f(x)$ with respect to $x$ is given by:

f'(x) = n \cdot x^{n-1}

In this case, we have $n = -6$ , so we have:

\frac{d}{dx} \left( \frac{1}{x^6} \right) = -6 \cdot \frac{1}{x^7}

Combining the Results

Now that we have found the derivatives of the logarithmic and rational terms, we can combine the results to find the derivative of the function $y = \frac{\ln x}{x^6}$ . We have:

\frac{dy}{dx} = -6 \cdot \frac{1}{x^7} \cdot \ln x + \frac{1}{x^6} \cdot \frac{1}{x}

Simplifying the Result

We can simplify the result by combining the terms:

\frac{dy}{dx} = -6 \cdot \frac{\ln x}{x^7} + \frac{1}{x^7}

Conclusion

In this article, we have differentiated the function $y = \frac{\ln x}{x^6}$ using the chain rule and the power rule of differentiation. We have found that the derivative of the function is given by:

\frac{dy}{dx} = -6 \cdot \frac{\ln x}{x^7} + \frac{1}{x^7}

This result can be used to find the rate of change of the function with respect to one of its variables.

Example Problems

Problem 1

Find the derivative of the function $y = \frac{\ln x}{x^4}$ .

Solution

To find the derivative of the function $y = \frac{\ln x}{x^4}$ , we can use the chain rule and the power rule of differentiation. We have:

y = \frac{\ln x}{x^4} = \frac{1}{x^4} \cdot \ln x

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

\frac{dy}{dx} = \frac{d}{dx} \left( \frac{1}{x^4} \right) \cdot \ln x + \frac{1}{x^4} \cdot \frac{d}{dx} (\ln x)

We can find the derivatives of the logarithmic and rational terms using the results from the previous section. We have:

\frac{d}{dx} (\ln x) = \frac{1}{x}

\frac{d}{dx} \left( \frac{1}{x^4} \right) = -4 \cdot \frac{1}{x^5}

Now, we can combine the results to find the derivative of the function $y = \frac{\ln x}{x^4}$ . We have:

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

Problem 2

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

Solution

To find the derivative of the function $y = \frac{\ln x}{x^3}$ , we can use the chain rule and the power rule of differentiation. We have:

y = \frac{\ln x}{x^3} = \frac{1}{x^3} \cdot \ln x

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

\frac{dy}{dx} = \frac{d}{dx} \left( \frac{1}{x^3} \right) \cdot \ln x + \frac{1}{x^3} \cdot \frac{d}{dx} (\ln x)

We can find the derivatives of the logarithmic and rational terms using the results from the previous section. We have:

\frac{d}{dx} (\ln x) = \frac{1}{x}

\frac{d}{dx} \left( \frac{1}{x^3} \right) = -3 \cdot \frac{1}{x^4}

Now, we can combine the results to find the derivative of the function $y = \frac{\ln x}{x^3}$ . We have:

\frac{dy}{dx} = -3 \cdot \frac{1}{x^4} \cdot \ln x + \frac{1}{x^3} \cdot \frac{1}{x}

Problem 3

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

Solution

To find the derivative of the function $y = \frac{\ln x}{x^2}$ , we can use the chain rule and the power rule of differentiation. We have:

y = \frac{\ln x}{x^2} = \frac{1}{x^2} \cdot \ln x

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

\frac{dy}{dx} = \frac{d}{dx} \left( \frac{1}{x^2} \right) \cdot \ln x + \frac{1}{x^2} \cdot \frac{d}{dx} (\ln x)

We can find the derivatives of the logarithmic and rational terms using the results from the previous section. We have:

\frac{d}{dx} (\ln x) = \frac{1}{x}

\frac{d}{dx} \left( \frac{1}{x^2} \right) = -2 \cdot \frac{1}{x^3}

Now, we can combine the results to find the derivative of the function $y = \frac{\ln x}{x^2}$ . We have:

\frac{dy}{dx} = -2 \cdot \frac{1}{x^3} \cdot \ln x + \frac{1}{x^2} \cdot \frac{1}{x}

Problem 4

Find the derivative of the function $y = \frac{\ln x}{x}$ .

Solution

To find the derivative of the function $y = \frac{\ln x}{x}$ , we can use the chain rule and the power rule of differentiation. We have:

y = \frac{\ln x}{x} = \frac{1<br/> **Q&A: Differentiate.$y = \frac{\ln x}{x^6}$** =====================================================

Q: What is the derivative of the function $y = \frac{\ln x}{x^6}$ ?

A: The derivative of the function $y = \frac{\ln x}{x^6}$ is given by: