Question

For the following exercises, compute \(d y / d x\) by differentiating \(\ln y\). $$ y=x^{-1 / \ln x} $$

Short answer

\( \frac{dy}{dx} = 0 \)

Step-by-step solution

Express the equation as a natural logarithm

We start by taking the natural logarithm of both sides of the equation. This results in: \[ \ln y = \ln(x^{-1/\ln x}) \]. Using the property of logarithms that \( \ln(a^b) = b \ln(a) \), this simplifies to \[ \ln y = -\frac{1}{\ln x} \cdot \ln x = -1. \] So, \( \ln y = -1 \).

Differentiate both sides with respect to x

We now differentiate both sides of the equation \( \ln y = -1 \) with respect to \( x \). The derivative of the left side is \( \frac{1}{y} \frac{dy}{dx} \). The derivative of the right side, which is a constant, is 0. Therefore, we have: \[ \frac{1}{y} \frac{dy}{dx} = 0. \]

Solve for \( \frac{dy}{dx} \)

To solve for \( \frac{dy}{dx} \), multiply both sides by \( y \): \[ \frac{dy}{dx} = 0 \cdot y = 0. \] Therefore, \( \frac{dy}{dx} = 0 \).

Key concepts

Natural Logarithm

The natural logarithm, commonly written as \( \ln(x) \), is the logarithm to the base \( e \), where \( e \) is an irrational and transcendental number approximately equal to 2.71828. It is fundamental in many areas of mathematics, particularly calculus. The natural logarithm has several unique properties that make it very versatile in differentiation and integration. Here are a few important properties:

• \( \ln(ab) = \ln(a) + \ln(b). \) This property allows us to transform a multiplication inside the logarithm into a sum of logarithms. It's incredibly useful for expanding expressions.
• \( \ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b). \) This property helps in turning divisions into differences of logarithms, making calculations simpler.
• \( \ln(a^b) = b\ln(a). \) As seen in the exercise, this property lets us bring down exponents, a key step in solving many logarithmic differentiation problems.

In the provided exercise, we used these properties to simplify the expression \( \ln(x^{-1/\ln x}) \) into \( -1 \). Understanding how to manipulate natural logarithms is essential for many calculus problems and sets the foundation for more complex calculations.

Derivative Properties

When using derivatives, it’s important to understand the key rules and properties that help simplify the process. Differentiation is the method by which we calculate the derivative of a function, which expresses how the function changes as its input changes. Here are some fundamental properties of derivatives:

• Power Rule: If \( f(x) = x^n \), then \( f'(x) = nx^{n-1} \). This rule is used frequently when dealing with polynomial expressions.
• Constant Rule: The derivative of a constant is zero. This was applied directly in the exercise when differentiating \( -1 \), resulting in \( 0 \).
• Chain Rule: If a function \( y \) is composed of another function, say \( f(g(x)) \), then the derivative \( \frac{dy}{dx} = f'(g(x))g'(x) \). This rule is vital for dealing with compositions of functions.

In the solution, after expressing \( \ln(y) = -1 \), differentiating both sides with respect to \( x \) utilized these simple rules, ultimately leading to the derivative \( \frac{dy}{dx} = 0 \). Mastery of these derivative properties ensures an easier path through calculus topics, as they apply to a wide range of functions.

Implicit Differentiation

Implicit differentiation is a technique used when the relationship between variables is given implicitly rather than explicitly. In this exercise, \( y \) is expressed using the function \( x \) in an indirect manner, which is typical in implicit differentiation. Here’s how it generally works:

• Differentiate Both Sides: When given an equation involving \( y \), differentiate both sides with respect to \( x \). This might involve using the chain rule, which considers \( y \) as a function of \( x \).
• Include \( \frac{dy}{dx} \): As part of the differentiation process, when you differentiate a term involving \( y \), make sure to append \( \frac{dy}{dx} \) to the result. This reflects the derivative of \( y \) with respect to \( x \).
• Solve for \( \frac{dy}{dx} \): Rearrange the equation to isolate \( \frac{dy}{dx} \) on one side. This gives the derivative of \( y \) in terms of \( x \).

In our example, after expressing the equation as \( \ln(y) = -1 \), implicit differentiation allowed us to find that \( \frac{dy}{dx} = 0 \) by differentiating both sides and solving for \( \frac{dy}{dx} \). Understanding implicit differentiation is crucial for tackling complex relationships that aren’t easily expressed as \( y = f(x) \).