Problem 387 - Calculus Volume 1

Question

The following exercises are intended to derive the fundamental properties of the natural log starting from the definition \(\ln (x)=\int_{1}^{x} \frac{d t}{t}, \quad\) using properties of the definite integral and making no further assumptions. Pretend, for the moment, that we do not know that \(e^{x}\) is the inverse function of \(\ln x,\) but keep in mind that \(\ln x\) has an inverse function defined on \((-\infty, \infty) .\) Call it \(E\) Show that \(E^{\prime}(t)=E(t).\)

Step-by-Step Solution

Answer

The derivative of \(E(t)\), \(E'(t)\), is equal to \(E(t)\).

Step 1: Understand the Problem

We need to demonstrate that the derivative of the inverse function of \(\ln x\), denoted as \(E(t)\), is equal to the function itself, \(E'(t) = E(t)\). To do this, we will utilize the relationship between derivatives and integrals, involving the definition of the natural log and its inverse.

Step 2: Differentiate the Exponential Function

Since \(\ln x\) and \(E(t)\) are inverse functions, we can express \(E(t)\) as the function that satisfies \(\ln E(t) = t\). Implicitly differentiate \(\ln E(t) = t\) with respect to \(t\).

Step 3: Apply Implicit Differentiation

Using implicit differentiation on \(\ln E(t) = t\), we get:\[ \frac{d}{dt}[\ln E(t)] = \frac{d}{dt}[t] = 1. \] The derivative of \(\ln E(t)\) is \(\frac{1}{E(t)} E'(t)\).

Step 4: Solve for the Derivative

From Step 3, we have:\[ \frac{1}{E(t)} E'(t) = 1. \] Multiply both sides by \(E(t)\) to isolate \(E'(t)\):\[ E'(t) = E(t). \] Thus, we have shown that the derivative of \(E(t)\) is equal to the function itself, i.e., \(E'(t) = E(t)\).

Key Concepts

Definite Integral

The concept of a definite integral is foundational in calculus. It represents the accumulation of quantities such as area under a curve within a specified interval. Consider the natural logarithm defined as \( \ln(x) = \int_{1}^{x} \frac{1}{t} \, dt \). This expression defines the natural log as the area under the curve \( y = \frac{1}{t} \) from \( t = 1 \) to \( t = x \).

• The upper and lower limits of the integral, \( x \) and \( 1 \), set the bounds for this accumulation of area.
• In this context, the integral is both a function itself and a means to define \( \ln(x) \).

Understanding definite integrals is crucial because they connect the geometry of a curve with its algebraic properties, allowing us to derive concepts like the natural logarithm from geometric principles.

Inverse Function

Inverse functions reverse the operations of a given function. For the function \( f(x) \), its inverse, denoted \( f^{-1}(y) \), satisfies \( f(f^{-1}(y)) = y \) and \( f^{-1}(f(x)) = x \). With the natural logarithm \( \ln(x) \) and its inverse \( E(x) \), we examine how these relationships apply.

• The definition \( \ln(E(x)) = x \) ensures that evaluating the inverse function and its original yields the starting value.
• This property confirms that the natural log and exponential functions "undo" each other.

Using \( E(t) \) allows us to explore the properties of natural logs without prior knowledge of the exponential function, focusing instead on the inherent behavior of inverse functions.

Implicit Differentiation

Implicit differentiation is a technique used when a relationship between variables isn't given explicitly, but rather implicitly. This means instead of \( y = f(x) \), we might have a relation like \( F(x, y) = 0 \) that doesn't solve for \( y \) directly. For \( E(t) \), the relationship \( \ln E(t) = t \) doesn't isolate \( E(t) \), but we can still differentiate it.

• By differentiating both sides with respect to \( t \), \( \frac{d}{dt} [\ln(E(t))] = \frac{d}{dt}[t] = 1 \), we employ implicit differentiation.
• This process results in \( \frac{1}{E(t)} E'(t) = 1 \), which isolates \( E'(t) \) upon simplification.

Implicit differentiation is powerful for deriving derivatives when functions are intertwined or not easily separated.

Exponential Function

The exponential function is a natural pair to the logarithmic function. While the logarithm deals with exponents and roots, the exponential function undoes these operations, returning a base raised to varying powers. If the natural logarithm \( \ln(x) \) is defined as a certain area under \( y = \frac{1}{t} \), its inverse, \( E(x) \), must exponentially grow.

• An important property of exponential functions is that the derivative \( E'(t) = E(t) \), implying self-similarity and consistent growth.
• This unique characteristic means that, unlike many other functions, exponential functions grow at a rate proportional to their current value.

Understanding these principles shows why the exponential function so naturally fits as the inverse of the logarithmic function, mirroring its properties in reverse.