Problem 84 - Calculus and its applications

Question

Differentiate. $$ f(x)=\ln \left(e^{x}-2\right) $$

Step-by-Step Solution

Answer

The derivative is \( f'(x) = \frac{e^x}{e^x - 2} \).

Step 1: Identify the Function Type

The function given is a composite function involving a natural logarithm and an exponential function. The outer function is the natural logarithm, \( \ln u \), and the inner function is \( u = e^x - 2 \).

Step 2: Apply the Chain Rule

To differentiate \( f(x) = \ln(e^x - 2) \), we use the chain rule, which states \( \frac{d}{dx} \ln(u) = \frac{1}{u} \cdot \frac{du}{dx} \). First, find the derivative of the inner function \( u = e^x - 2 \).

Step 3: Differentiate the Inner Function

The inner function \( u = e^x - 2 \) differentiates to \( \frac{du}{dx} = \frac{d}{dx}(e^x - 2) = e^x \), since the derivative of \( e^x \) is \( e^x \) and the derivative of a constant is zero.

Step 4: Differentiate the Outer Function

Substitute \( u = e^x - 2 \) and \( \frac{du}{dx} = e^x \) into the chain rule relation, giving us \( f'(x) = \frac{1}{e^x - 2} \, \cdot \, e^x \).

Step 5: Simplify the Expression

The expression \( f'(x) = \frac{e^x}{e^x - 2} \) simplifies to this form directly upon substituting, as there's no further reduction needed.

Key Concepts

Chain Rule

The Chain Rule is a fundamental concept in calculus used to differentiate composite functions. Imagine a composite function as one function nested inside another, much like a matryoshka doll. To differentiate such a function, the Chain Rule allows us to break down the process into manageable steps. We apply it by taking the derivative of the outer function first, keeping the inner function untouched. After this, we multiply this result by the derivative of the inner function. In mathematical terms, if you have a function, say \( f(g(x)) \), the derivative \( f'(g(x)) \) is calculated as \( f'(g(x)) \times g'(x) \). This rule is extremely handy when dealing with functions like natural logarithms and exponentials which frequently appear together in calculus problems.

Composite Function

A composite function is formed when one function is applied to the result of another function. It can be written as \( (f \circ g)(x) = f(g(x)) \), which means you first apply the function \( g \) on \( x \), and then the function \( f \) on \( g(x) \). In the exercise, we deal with a composite function \( \ln(e^x-2) \), where the natural logarithm function ln is applied to the result of \( e^x-2 \). Understanding the components of a composite function is crucial before applying differentiation techniques like the Chain Rule because each part must be addressed separately and precisely. With composite functions, each layer must be peeled back in a logical manner to reveal the underlying structure.

Natural Logarithm

The natural logarithm, denoted as \( \ln(x) \), is a logarithm with base \( e \), where \( e \approx 2.71828 \). This function is the inverse of the exponential function with base \( e \). The derivative of the natural logarithm function \( \ln(u) \) with respect to \( u \) is \( \frac{1}{u} \). In our exercise, \( \ln(e^x-2) \), \( e^x-2 \) acts as the argument of the logarithm. When differentiating this function with respect to \( x \), the Chain Rule is used, transforming the process into \( \frac{1}{e^x-2} \times e^x \). Grasping how the natural logarithm operates is essential for smoothly progressing through problems involving derivatives of composite functions.

Exponential Function

Exponential functions take the general form \( a^x \), but the most important is \( e^x \), where the base \( e \approx 2.71828 \) is a fundamental constant. This function profoundly impacts many fields from calculus to real-world models of growth. The derivative of \( e^x \) is particularly noteworthy because it remains \( e^x \); this unique property simplifies differentiation significantly. In the exercise, \( e^x \) forms part of the inner function \( u = e^x - 2 \). When differentiating \( e^x \), the simplicity of its derivative allows for straightforward application in composite functions, requiring no further adjustment to its basic form. Mastering the exponential function's behavior is key for effectively handling complex mathematical models.