Problem 11 - Calculus and its applications

Question

Differentiate. $$ f(x)=e^{x} $$

Step-by-Step Solution

Answer

The derivative of \( f(x) = e^{x} \) is \( f'(x) = e^{x} \).

Step 1: Identify the Function Type

The function we are working with is an exponential function, specifically, the natural exponential function, represented as \( f(x) = e^{x} \). The base is the constant \( e \), which is approximately equal to 2.71828.

Step 2: Recall the Derivative Rule for Exponential Functions

The derivative of exponential functions of the form \( f(x) = a^{x} \), where \( a \) is a constant, is given by \( f'(x) = a^{x} \, \ln(a) \). In the case of the natural exponential function where \( a = e \), the derivative simplifies to \( f'(x) = e^{x} \ln(e) \).

Step 3: Simplify Using Properties of \( e \)

Recall that the natural logarithm of \( e \), denoted as \( \ln(e) \), is equal to 1. Therefore, the expression \( f'(x) = e^{x} \ln(e) \) simplifies further to \( f'(x) = e^{x} \cdot 1 = e^{x} \).

Step 4: Write the Final Derivative

We conclude that the derivative of the function \( f(x) = e^{x} \) is simply the original function itself, which is \( f'(x) = e^{x} \).

Key Concepts

Differentiation rules

Differentiation is the process of finding the derivative of a function. It provides information about the rate of change or slope of the function. Understanding differentiation rules is crucial because these rules help us find derivatives quickly and efficiently.

In calculus, several basic differentiation rules serve as building blocks for other complex functions. Some of these basic rules include:

• Constant Rule: The derivative of a constant is zero.
• Power Rule: For a function of the form \( f(x) = x^n \), where \( n \) is a constant, the derivative is \( f'(x) = nx^{n-1} \).
• Sum Rule: The derivative of the sum of two functions is the sum of their derivatives.
• Product Rule: If two functions are multiplied together, their derivative is \( (uv)' = u'v + uv' \).
• Quotient Rule: If one function is divided by another, their derivative is \( \left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2} \).
• Chain Rule: Used when dealing with composite functions, the derivative is \( (f(g(x)))' = f'(g(x))g'(x) \).

By mastering these rules, you can tackle a wide variety of differentiation problems, including the differentiation of exponential functions. Differentiation is not just about following rules but also understanding the underlying concepts and applications.

Exponential functions

Exponential functions are a special type of mathematical functions where a constant base is raised to a variable exponent. They are typically written in the form \( f(x) = a^x \), where \( a \) is a constant greater than zero and \( x \) is the exponent.

One of the most famous exponential functions is the natural exponential function, \( f(x) = e^x \), where \( e \) is the base of the natural logarithm, approximately equal to 2.71828. The beauty of exponential functions, especially those with base \( e \), is their simplicity when it comes to differentiation.

Exponential functions have unique properties:

• Rapid Growth: Exponential functions grow or decay at a rate that is proportional to their value.
• Smooth Curves: They are continuous and have smooth curves without breaks.
• Derivative Equal to Itself: For the natural exponential function \( e^x \), the derivative is \( e^x \). In other words, it remains unchanged after differentiation.

These properties make exponential functions particularly important in fields ranging from finance to natural sciences, where growth processes often follow exponential patterns.

Natural logarithm properties

The natural logarithm, denoted as \( \ln(x) \), is the logarithm to the base \( e \). It is a fundamental concept in calculus and is often used when working with exponential functions.

Understanding the properties of natural logarithms helps simplify complex mathematical equations and aids in the differentiation process.

Here are some key properties of the natural logarithm:

• Logarithm of \( e \): The natural logarithm of the base \( e \) is 1, so \( \ln(e) = 1 \).
• Product Property: \( \ln(xy) = \ln(x) + \ln(y) \).
• Quotient Property: \( \ln\left(\frac{x}{y}\right) = \ln(x) - \ln(y) \).
• Power Property: \( \ln(x^n) = n \ln(x) \).

These properties make natural logarithms very useful for solving equations involving exponential growth and decay. When differentiating exponential functions, particularly \( e^x \), these properties simplify the process significantly, as seen in how \( \ln(e) = 1 \) transforms the derivative of \( e^x \) to simply \( e^x \). By mastering these principles, you can better handle all sorts of exponential and logarithmic differentiations.