Problem 72 - Calculus and its applications

Question

Differentiate. $$ y=\frac{\ln x}{x^{4}} $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Type of Function

The given function is a quotient, i.e., a function divided by another function: \( y = \frac{\ln x}{x^4} \). To differentiate it, we'll use the quotient rule for derivatives.

Step 2: Write Down the Quotient Rule

The quotient rule states that if you have a function \( y = \frac{u}{v} \), the derivative is \( \frac{dy}{dx} = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2} \). Here, \( u = \ln x \) and \( v = x^4 \).

Step 3: Differentiate the Numerator

Differentiate \( u = \ln x \) with respect to \( x \). The derivative \( \frac{du}{dx} = \frac{1}{x} \) because the derivative of \( \ln x \) is \( \frac{1}{x} \).

Step 4: Differentiate the Denominator

Differentiate \( v = x^4 \) with respect to \( x \). The derivative \( \frac{dv}{dx} = 4x^3 \) because of the power rule \( \frac{d}{dx} (x^n) = nx^{n-1} \).

Step 5: Apply the Quotient Rule

Apply the quotient rule formula: \( \frac{dy}{dx} = \frac{x^4 \cdot \frac{1}{x} - \ln x \cdot 4x^3}{(x^4)^2} \).

Step 6: Simplify the Derivative

Simplify the expression: \( \frac{dy}{dx} = \frac{x^3 - 4x^3 \ln x}{x^8} \). Factor \( x^3 \) from the numerator: \( \frac{dy}{dx} = \frac{x^3 (1 - 4 \ln x)}{x^8} \). Further simplify to \( \frac{dy}{dx} = \frac{1 - 4 \ln x}{x^5} \).

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus, representing the process of finding the derivative of a function.
The derivative of a function describes how the function value changes as its input changes. In simpler terms, it gives us the slope of the tangent line at any point on the graph of the function.
There are several differentiation rules, such as the power rule, product rule, and quotient rule, which facilitate the calculation of derivatives.

• The power rule is used when differentiating expressions involving powers of a variable. It's stated as: if \( f(x) = x^n \), then \( f'(x) = nx^{n-1} \).


• For additions or subtractions of functions, you differentiate each term separately.


• The quotient rule, like in this exercise, is necessary when you have one function dividing another.

Differentiation helps in optimizing problems, analyzing motion, calculating rates of change, and modeling real-world phenomena.

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions. They help to undo the exponential functions like \( e^x \). The natural logarithm, denoted as \( \ln x \), is a special kind of logarithm because it's based on the constant \( e \), approximately equal to 2.718.
The function \( \ln x \) is very handy when you're dealing with growth processes, compounding problems, or transformations, thanks to its unique properties.
When differentiating a logarithmic function, there is a specific rule you need to remember:

• The derivative of \( \ln x \) is \( \frac{1}{x} \).

This tells us how rapidly \( \ln x \) increases as \( x \) increases.
Logarithms are widely used in various fields, from solving equations to enabling easier multiplication and division through their properties, which transform them into addition and subtraction, respectively.

Power Rule

The power rule is a straightforward and highly efficient way to differentiate functions where the variable is raised to a power.
It simplifies differentiation whenever we encounter terms of the form \( x^n \).
When you apply the power rule:

• Start with a function \( f(x) = x^n \).


• The derivative, \( f'(x) \), becomes \( nx^{n-1} \).

This rule is fundamental in handling polynomial expressions and can be applied multiple times in more complex functions comprising several terms. For instance, in the denominator of our exercise function \( x^4 \), applying the power rule helps find \( 4x^3 \) as the derivative. This simplifies many calculus problems where complex expressions could otherwise be cumbersome to differentiate.
By mastering the power rule, you'll efficiently handle and simplify derivatives of polynomial functions, making it a core tool for calculus students.