Problem 60 - Calculus - AP Edition

Question

Review the derivation of the formula $$\frac{d}{d x}[\ln x]=\frac{1}{x}$$ and then write a paragraph that discusses all the ingredients (theorems, limit properties, etc.) that are needed for this derivation.

Step-by-Step Solution

Answer

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

Step 1: Understanding the Function

The function we are dealing with is the natural logarithm function, \( \ln x \). It is defined for \( x > 0 \). We are interested in finding the derivative of this function with respect to \( x \).

Step 2: Recall the Derivative Definition

The derivative \( \frac{d}{dx}[f(x)] \) of a function \( f(x) \) is defined as the limit \[ \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \]. We will use this definition to find \( \frac{d}{dx}[\ln x] \).

Step 3: Apply the Derivative Definition to \( \ln x \)

Substitute \( f(x) = \ln x \) into the definition of the derivative to get \[ \lim_{h \to 0} \frac{\ln(x+h) - \ln x}{h} \]. To simplify this, we use the logarithmic identity \( \ln a - \ln b = \ln \left( \frac{a}{b} \right) \).

Step 4: Simplify Using Logarithmic Properties

The expression \( \ln(x+h) - \ln x \) becomes \( \ln \left( \frac{x+h}{x} \right) \). So the limit becomes \[ \lim_{h \to 0} \frac{\ln \left( 1 + \frac{h}{x} \right)}{h} \].

Step 5: Use Limit and Logarithm Properties

Now, we recognize that as \( h \to 0 \), \( \frac{h}{x} \to 0 \). We use the first principle of limits for the logarithm, specifically that \( \ln(1 + u) \approx u \) when \( u \) is small, which transforms the limit into \[ \lim_{h \to 0} \frac{\frac{h}{x}}{h} = \lim_{h \to 0} \frac{1}{x} \].

Step 6: Final Simplification

The limit simplifies to \( \frac{1}{x} \) since \( h \) cancels out in the numerator and denominator. Thus, \( \frac{d}{dx}[\ln x] = \frac{1}{x} \).

Key Concepts

Limits in Calculus

Limits in calculus are fundamental tools for understanding the behavior of functions as they approach particular points. In our exercise, the concept of a limit is used to find the derivative of the natural logarithm function, \[ \ln x \].
We apply the definition of the derivative, which relies heavily on limits. This definition states that the derivative of a function \( f(x) \) at a point \( x \) is the limit of the rate of change of \( f(x) \) as it approaches that point.
The limit is formally given by \[ \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \].
When we substitute \( f(x) = \ln x \) into this formula, we embark on a journey of simplifying the expression using logarithmic properties.

• The fundamental idea is to explore how \( \ln x \) behaves as it changes infinitesimally—that is, as \( h \) approaches zero.
• The concept of limits also embraces the approximation \( \ln(1 + u) \approx u \), useful when \( u \) is very small.

The mastery of limits allows us to process changes in \( x \) effectively, ultimately helping to derive consistent results for the behavior of the function.

Logarithmic Differentiation

Logarithmic differentiation is a powerful technique for finding the derivatives of functions involving logarithms. Here, we applied it to the natural logarithm \( \ln x \). It's particularly useful when dealing with complex products, quotients, or other functions that naturally involve logarithms.
When differentiating \( \ln x \), the key property utilized is that of the difference of logs: \( \ln a - \ln b = \ln \left( \frac{a}{b} \right) \). This property simplifies expressions inside the derivative formula.

• In our case, \( \ln(x+h) - \ln x \) simplifies to \( \ln \left( \frac{x+h}{x} \right) \).
• This transformation aids in analyzing how small changes in \( x \) affect the natural log.

The benefit of logarithmic differentiation lies in its ability to reduce complexities inherent in manipulating logarithmic functions during derivation. Understanding these manipulations is pivotal when calculating \( f(x) \) derivatives where logarithmic identities come into play naturally.

Definition of Derivative

The definition of a derivative is a cornerstone of calculus, representing the fundamental idea of rate of change or slope of a function at a point. To derive \( \frac{d}{dx}[\ln x] \), we start with this definition.
The derivative of a function \( f(x) \) is defined as the limit\[ \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \].
This expression calculates the slope of the tangent line to the function at a point.

• For \( \ln x \), we put \( f(x) = \ln(x) \) into this definition.
• This allows us to analyze the change in \( \ln x \) as \( x \) changes by a small amount \( h \).

The subsequent simplification using \( \ln \) properties helps transform this definition into something tangible. Recognizing how derivatives represent instantaneous rates of change makes them a vital tool for exploring function behaviors dynamically. When \( h \) cancels out, as it does in the final steps, we get the clean result of \( \frac{1}{x} \), emphasizing the power and elegance of the derivative concept.