Problem 387 - Calculus Volume 1

Question

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 \(E(t)\) is equal to itself: \(E'(t) = E(t)\).

Step 1: Problem Introduction

We want to demonstrate that the derivative of the inverse function of \(\ln x\), denoted as \(E\), is equal to the function itself. So we need to show \(E'(t) = E(t)\).

Step 2: Understand the Inverse Relationship

By definition, \(E(t)\) implies \(\ln(E(t)) = t\). This means \(E\) is the inverse of \(\ln\) and for any \(t\), applying \(\ln\) on \(E(t)\) should give back \(t\).

Step 3: Implicit Differentiation

Differentiate both sides of the equation \(\ln(E(t)) = t\) with respect to \(t\). This yields \(\frac{d}{dt}[\ln(E(t))] = \frac{d}{dt}[t]\).

Step 4: Apply Chain Rule

Using the chain rule, the left side becomes \(\frac{1}{E(t)} \cdot E'(t)\), while the derivative of \(t\) is 1. Thus, \(\frac{1}{E(t)} \cdot E'(t) = 1\).

Step 5: Solve for \(E'(t)\)

Multiply both sides of the equation \(\frac{1}{E(t)} \cdot E'(t) = 1\) by \(E(t)\) to isolate \(E'(t)\). This results in \(E'(t) = E(t)\).

Step 6: Conclusion

We have shown that the derivative of \(E(t)\), which is the inverse function of \(\ln x\), is equal to the function itself. Hence, \(E'(t) = E(t)\).

Key Concepts

Differentiation

In mathematics, differentiation is a fundamental concept that deals with how a function changes as its input changes. Essentially, it answers the question of how a quantity varies with respect to another variable. When we differentiate a function, we find its derivative.

• Think of the derivative as the rate of change or slope of the function at a given point.
• If you picture a curve on a graph, the derivative is the slope of the tangent line at any specific point on the curve.

In the original exercise, we differentiate to explore the behavior of the inverse function of the natural logarithm, denoted as \(E(t)\). In step 3 of the solution, we used this concept to differentiate both sides of the equation \(\ln(E(t)) = t\). Differentiating the left side involves the use of logarithmic differentiation, while the right side, \(t\), simply becomes 1 due to the basic rule of differentiation for linear functions. Understanding how to perform these operations enables us to reveal the relationship between the function and its derivative.

Natural Logarithm

The natural logarithm, symbolized by \(\ln(x)\), is a logarithm to the base \(e\), where \(e\) is an irrational number approximately equal to 2.71828. It is one of the most important functions in calculus, particularly because it describes continuous growth processes.

• The function \(\ln(x)\) describes the time required for continuous growth to reach a given amount.
• It is the inverse of the exponential function \(e^x\).

In this exercise, we examined \(\ln x\) to establish that it has an inverse function, called \(E\), demonstrating that \(\ln(E(t)) = t\). This means that applying \(\ln\) to \(E(t)\) returns \(t\), a fundamental property of inverse functions. This inverse relationship is crucial for understanding natural logarithms in calculus and its applications.

Chain Rule

The chain rule is an essential tool in calculus for finding the derivative of composite functions. A composite function is like a function within a function, such as \(\ln(E(t))\) where one function is nested inside another.

• The chain rule expresses that the derivative of a composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function.
• In symbolic terms, if \(y = f(g(t))\), then \(\frac{dy}{dt} = f'(g(t)) \cdot g'(t)\).

In the solution we employed this rule during Step 4, where we differentiated \(\ln(E(t))\) with respect to \(t\). The chain rule helps us convert the differentiation of nested functions into something manageable by breaking it down into steps. Here, the outer function is \(\ln(x)\), whose derivative is \(\frac{1}{x}\), and the inner function is \(E(t)\). Thus, the derivative is \(\frac{1}{E(t)} \cdot E'(t)\). Understanding the chain rule is vital for solving complex problems involving layers of functions as it simplifies the differentiation process.