Problem 10 - Calculus Single Variable

Question

An expression for \(f(x)\) is given. Compute the first, second, and third derivatives of \(f(x)\) with respect to \(x\). \(\exp (-x)\)

Step-by-Step Solution

Answer

The derivatives are: \( f'(x) = -e^{-x} \), \( f''(x) = e^{-x} \), \( f'''(x) = -e^{-x} \).

Step 1: Understanding the Function

The function given is \( f(x) = \exp(-x) \), which is another way to write the exponential function \( e^{-x} \). We'll find its derivatives with respect to \( x \).

Step 2: Finding the First Derivative

To find the first derivative of \( f(x) = e^{-x} \), apply the rule for derivatives of exponential functions. The derivative of \( e^{-x} \) with respect to \( x \) is \( -e^{-x} \). Thus, \( f'(x) = -e^{-x} \).

Step 3: Finding the Second Derivative

Now, take the derivative of the first derivative \( f'(x) = -e^{-x} \). The derivative of \( -e^{-x} \) with respect to \( x \) is \( e^{-x} \). Thus, \( f''(x) = e^{-x} \).

Step 4: Finding the Third Derivative

Take the derivative of the second derivative \( f''(x) = e^{-x} \). The derivative of \( e^{-x} \) with respect to \( x \) is \( -e^{-x} \). Thus, \( f'''(x) = -e^{-x} \).

Key Concepts

Exponential Functions

Exponential functions are incredibly important in mathematics. They are functions in the form of \( f(x) = e^{x} \), where \( e \) is a mathematical constant approximately equal to 2.71828. This constant is known as Euler's number, and is the base of natural logarithms.
The function \( e^{x} \) grows very rapidly as \( x \) increases. However, when we work with negative exponents, like \( e^{-x} \), the function decreases as \( x \) increases.
In general, exponential functions have the following characteristics:

• They are continuous and smooth across their entire domain.
• They never touch the x-axis, meaning they are always positive (except for transformations like negative exponents).

Understanding how to work with exponential functions is crucial when learning calculus, especially for solving differential equations and modeling growth or decay in real-world problems.

Calculus

Calculus is a branch of mathematics focused on change. It deals with how quantities change and allows us to compute things like slopes, areas under curves, and volumes.
It is divided into two main areas: differential calculus and integral calculus. This exercise falls under differential calculus, which is concerned with the concept of a derivative.
A derivative represents the rate of change of a function with respect to a variable. It's a bit like finding the slope of a curve at a specific point. In real-life scenarios, this could mean finding the velocity of an object at a particular time if you know how its position changes over time.
The notation for a derivative of a function \( y = f(x) \) with respect to \( x \) is quite flexible and includes \( f'(x) \) or \( \frac{dy}{dx} \). These symbols help communicate the mathematics behind the changes we are examining.

Differentiation Rules

Differentiation is the process of finding a derivative. There are several rules that make this process easier. Understanding these rules simplifies the computation of derivatives, especially in complex functions.
The first important rule is the "constant multiple rule." This says if a function \( f(x) \) is multiplied by a constant, the derivative of this product is the constant times the derivative of the function.
Another vital rule is the "exponential function rule." For a function of form \( f(x) = e^{g(x)} \), the derivative is \( f'(x) = e^{g(x)} \cdot g'(x) \). This rule helps us compute derivatives for exponential functions, like in our exercise.
When finding derivatives, knowing these rules speeds up the process. It means we can focus less on the mechanics and more on understanding what the derivative tells us about the function, like its increasing/decreasing behavior. These rules form the backbone of much of what we do in calculus, aiding in the problem-solving process.