Question

In Exercises, find the derivative of the function. $$ g(x)=e^{-x} \ln x $$

Short answer

The derivative of the function \(g(x) = e^{-x} \ln x\) is \(g'(x) = -e^{-x} \cdot \ln x + \frac{e^{-x}}{x}\)

Step-by-step solution

Identify the functions

The first function \(f(x) = e^{-x}\) and the second function \(h(x) = \ln x\).

Compute the derivatives of the individual functions

The derivative of \(f(x) = e^{-x}\) is \(f'(x) = -e^{-x}\). The derivative of \(h(x) = \ln x\) is \(h'(x) = 1/x\).

Apply the Product Rule

According to the Product Rule, the derivative of \(g(x) = f(x) \cdot h(x)\) is given by \(g'(x) = f'(x) \cdot h(x) + f(x) \cdot h'(x)\). Substituting \(f(x)\), \(h(x)\), \(f'(x)\), and \(h'(x)\) into the formula, we get \(g'(x) = -e^{-x} \cdot \ln x + e^{-x} \cdot (1/x)\)

Key concepts

Derivative

A derivative measures how a function's output changes as its input changes. Imagine it as the formula for finding the slope of the function graph at any point. Derivatives allow us to understand the rate of change of a quantity that depends on another. In the context of the function \( g(x) = e^{-x} \ln x \), calculating its derivative requires understanding how each part of the function changes. Very often we use rules, such as the Product Rule, to simplify this process. The Product Rule is ideal when you have two multiplied functions, like in \( g(x) = f(x) \cdot h(x) \). To find the derivative, you'll compute the derivative of each part separately and combine them as per the rule: \( g'(x) = f'(x) \cdot h(x) + f(x) \cdot h'(x) \). This method helps break down complex derivatives into simpler parts.

Exponential Functions

Exponential functions are a category of functions that involve the exponent \(e\), where \(e\) is an important constant approximately equal to 2.71828. They are characterized by growth or decay processes. In the function \( f(x) = e^{-x} \), the negative exponent indicates a decay pattern rather than growth. The derivative of exponential functions maintains the form of the original function, which becomes evident when differentiating these types of functions. For \( e^{-x} \), the derivative is \( -e^{-x} \), showing the direct impact of the chain rule because of the negative exponent. Exponential functions are fundamental in various fields such as biology for modeling population growth or decay of radioactive substances.

Natural Logarithm

The natural logarithm, denoted as \( \ln x \), is the inverse of the exponential function with base \(e\). Simply put, it's asking the question: "To what power do we raise \(e\) to get \(x\)?" Logarithms transform multiplication into addition, making them incredibly useful in simplifying complex calculations. When you differentiate the natural logarithm \( \ln x \), you obtain \( \frac{1}{x} \), which is a fundamental result in calculus. This derivative is essential in solving integrals and finding slopes of logarithmic functions. Natural logarithms are invaluable in fields like finance for calculating compound interest and in engineering for measuring sound intensity.