Problem 67 - University Calculus: Early Transcendentals

Question

Find the derivative of \(y\) with respect to the given independent variable. $$y=2^{x}$$

Step-by-Step Solution

Answer

The derivative is \(\frac{dy}{dx} = 2^x \ln(2)\).

Step 1: Identify the Function

Given function is \(y = 2^x\). Here, \(2^x\) is an exponential function with base 2 and exponent \(x\).

Step 2: Recall the Derivative Formula for the Exponential Function

The formula for the derivative of an exponential function with base \(a\) is \(\frac{d}{dx} (a^x) = a^x \ln(a)\), where \(\ln\) denotes the natural logarithm.

Step 3: Differentiate Using the Formula

Applying the formula \(\frac{d}{dx} (a^x) = a^x \ln(a)\), with \(a = 2\), the derivative of \(y = 2^x\) with respect to \(x\) is: \[ \frac{dy}{dx} = 2^x \ln(2) \].

Key Concepts

Exponential Function

Exponential functions are prevalent in mathematics due to their unique properties and applications. An exponential function is defined as a function of the form \(y = a^x\), where \(a\) is a constant known as the base, and \(x\) is the exponent or power. These functions are characterized by their rapid growth or decay, depending on whether \(a\) is greater or less than 1.

For example, in the exercise, the function \(y = 2^x\) represents an exponential function with base \(2\). The base determines the rate at which the function increases. As \(x\) increases, the value of \(2^x\) grows much faster compared to linear or polynomial functions.

Exponential functions commonly appear in various real-world contexts like population growth, radioactive decay, and compound interest. Their exponential nature makes them incredibly useful for modeling processes that have a constant multiplicative rate of change. Understanding the differentiation of exponential functions is vital as it allows us to analyze how these functions change over their domain.

Natural Logarithm

The natural logarithm is a special type of logarithm with the base \(e\), an irrational number approximately equal to 2.71828. The natural logarithm is denoted as \(\ln(x)\). It is the inverse operation of raising \(e\) to a power.

In calculus, the natural logarithm is especially important because it serves as the derivative of the exponential function \(e^x\). This relationship simplifies many differential equations and makes the natural logarithm crucial in differentiation formulas for exponential functions with different bases.

When differentiating an exponential function like \(a^x\), where \(a\) is any positive number, the formula \(\frac{d}{dx} (a^x) = a^x \ln(a)\) is applied. Here, the \(\ln(a)\) emerges naturally and represents the instantaneous rate of change of the exponential function with respect to its base.

The natural logarithm helps convert multiplicative problems into additive ones through its properties, making it easier to work with problems involving exponential growth or decay.

Differentiation Formula

Differentiation is a core concept in calculus dealing with the rate at which a function changes. In simpler terms, it is a way of finding the slope or gradient of a curve at any given point. Differentiating functions allows you to understand their behavior and calculate how they react to changes in input.

The differentiation formula is an essential tool for finding the derivative of a function. For exponential functions with a given base \(a\), the derivative is determined using the formula: \[\frac{d}{dx} (a^x) = a^x \ln(a)\].

This differentiation formula shows that the derivative of \(a^x\) is the product of the original function \(a^x\) and the natural logarithm of the base \(a\).

• First, identify the base and the exponent of the function.
• Apply the formula, substituting the base of the function into the natural logarithm.
• This results in a new expression representing the derivative.

In the specific exercise, substituting \(2\) for \(a\) in our formula results in the derivative \(\frac{dy}{dx} = 2^x \ln(2)\). This allows students to find the rate of change of the function \(y = 2^x\) with respect to \(x\), providing insight into the function's behavior as \(x\) changes.