Problem 31 - Calculus Single Variable

Question

Calculate the derivative of the given xpression with respect to \(x\). $$ 2^{x} $$

Step-by-Step Solution

Answer

The derivative of \(2^x\) is \(2^x \cdot \ln(2)\).

Step 1: Understanding Exponential Derivatives

The derivative of an exponential function with a base other than 'e' requires the use of the chain rule. Recall, for the function of the form \(a^x\), the derivative is \(a^x \cdot \ln(a)\). In our case, the base \(a\) is 2.

Step 2: Identify the Base and Apply the Derivative Rule

Identify that the function \(2^x\) fits the form \(a^x\) where \(a = 2\). Using the derivative rule for exponential functions, the derivative is smaller parts to easier understand: \( \frac{d}{dx} [a^x] = a^x \cdot \ln(a) \).

Step 3: Calculate the Derivative

Apply the rule: the derivative of \(2^x\) is \(2^x \cdot \ln(2)\). This result is derived directly from applying the exponential derivative rule to our specific base (2).

Key Concepts

Chain Rule

The Chain Rule is a fundamental concept in calculus used for finding the derivative of compositions of functions. Whenever you have a function inside another function, the Chain Rule is your go-to tool. In our case with the exponential function, the base remains constant, but the exponent (x) acts as the inner function. Therefore, even though the Chain Rule might not explicitly look like it's used in standard exponential derivatives, it is intrinsically woven in the process.

To apply the Chain Rule, if you have a function of the form \(g(f(x))\), the derivative is computed as \(g'(f(x)) \cdot f'(x)\). For exponential functions like \(2^x\), the Chain Rule is applied because the function structure involves the inner function \(f(x) = x\) and outer function \(g(y) = 2^y\). Even if it isn't directly visible, this rule is part of the concept that underlies our approach and calculations.

Derivative Calculation

Derivative calculation is the process of finding the rate of change of a function with respect to a variable. For any function \(f(x)\), the derivative \(f'(x)\) tells us how much \(f(x)\) changes as \(x\) changes. The mechanics of calculating derivatives can vary greatly between different types of functions.

In the case of an exponential function like \(2^x\), we use a special rule: the derivative of \(a^x\) is \(a^x \cdot \ln(a)\). This means for \(2^x\), you plug in \(a = 2\), resulting in \(2^x \cdot \ln(2)\). The natural logarithm, \(\ln(2)\), is a constant that reflects the rate of growth of the base number 2. This derivative calculation becomes particularly useful when analyzing growth models or other exponential processes.

Exponential Functions

Exponential Functions are a key class of functions in mathematics, often characterized by a constant base raised to a variable exponent. They are expressed in the form \(a^x\), where \(a\) is a positive constant and \(x\) is the exponent. Exponential functions are known for their rapid growth or decay properties, depending on the base and the context.

In the derivative context, an important characteristic is that the rate of change of the function is proportional to the value of the function itself. This forms the basis for the exponential derivative rule. Whether you are dealing with growth processes like compound interest or decay processes like radioactive decay, exponential functions provide a powerful framework.

• An exponential function with base 2, \(2^x\), multiplies by 2 for each increase in \(x\).
• These functions are continuously increasing or decreasing, with no turning points or local extrema.

The properties of exponential functions make them prevalent in fields such as biology, finance, and physics.