Problem 33 - Calculus for Biology and Medicine

Question

Differentiate the functions in Problems 1-52 with respect to the independent variable. $$ f(x)=2^{x} $$

Step-by-Step Solution

Answer

The derivative is \\( f'(x) = 2^x \ln(2) \\).

Step 1: Definition of Differentiation

The derivative \( f'(x) \) of a function \( f(x) \) represents the rate at which \( f(x) \) changes with respect to the independent variable \( x \). For the function \( f(x) = 2^x \), we aim to find \( f'(x) \).

Step 2: Use the Exponential Rule

For the function \( a^x \), where \( a \) is a constant, the derivative \( \frac{d}{dx} [a^x] = a^x \ln(a) \). Apply this rule to the function \( f(x) = 2^x \).

Step 3: Apply the Exponential Rule

Using the rule from Step 2, substituting \( a = 2 \), we get \[ f'(x) = 2^x \ln(2) \].

Step 4: Simplified Derivative

The expression \( f'(x) = 2^x \ln(2) \) is the derivative of \( f(x) = 2^x \) with respect to \( x \). There is no further simplification needed for this expression.

Key Concepts

Understanding Exponential Functions

Exponential functions are a vital part of calculus and appear frequently in various mathematical contexts. They are functions of the form \( f(x) = a^x \), where \( a \) is a constant known as the base, and \( x \) is the exponent or power. A unique feature of exponential functions is that they grow at an increasing rate as the value of \( x \) increases. This is because the function's rate of change is directly proportional to the value of the function itself.

In real-world scenarios, exponential functions can model phenomena that grow quickly over time, such as population growth, radioactive decay, and compound interest. Recognizing the structure and behavior of exponential functions is crucial for understanding how to differentiate them.

The exponential function \( f(x) = 2^x \) specifically involves a base of 2. This means that as \( x \) increases, \( 2^x \) will double with each increment of 1 in \( x \), showing how rapidly these functions can grow. Learning how to differentiate them is an essential skill in calculus.

Exploring Derivative Rules

Differentiation is a core concept in calculus that involves finding the derivative of a function. The derivative represents how a function changes with respect to its independent variable. Several rules are used to differentiate functions depending on their form.

For exponential functions like \( a^x \), the differentiation rule is:

• \( \frac{d}{dx}[a^x] = a^x \ln(a) \)

This rule tells us how to find the rate of change of an exponential function. It combines the original function with the natural logarithm of its base, \( a \).

Other differentiation rules include:

• The Power Rule: Used for functions like \( x^n \)
• The Product Rule: For the product of two functions
• The Quotient Rule: For the quotient of two functions
• The Chain Rule: For composite functions

Mastering these rules is crucial because they provide the necessary tools to tackle different kinds of differentiation problems.

Effective Calculus Problem Solving

Solving calculus problems, especially those involving differentiation, requires a structured approach. Here’s an effective process:

1. **Identify the Function Type:** Recognize whether you are dealing with an exponential, polynomial, trigonometric, or logarithmic function. This understanding dictates which derivative rules to apply.

2. **Choose the Appropriate Derivative Rule:** Depending on the function's structure, select the relevant rule, such as the exponential rule for exponential functions.

3. **Apply the Rule Carefully:** Substitute known values into the differentiation formula. For example, in differentiating \( f(x) = 2^x \), use the rule \( \frac{d}{dx}[2^x] = 2^x \ln(2) \).

4. **Simplify if Necessary:** After differentiation, simplify the expression, if possible. However, some derivatives, like \( f'(x) = 2^x \ln(2) \), are already in a simplified form.

5. **Verify Your Result:** Make sure that your derivative makes sense by considering the context of the problem or using derivative calculators to validate your work.

By following these steps, you can solve calculus problems more effectively and deepen your conceptual understanding of differentiation.