Problem 39 - College Algebra and Calculus: An Applied Approach

Question

In Exercises, find the derivative of the function. $$ y=3^{x} $$

Step-by-Step Solution

Answer

The derivative, \( \dot{y} = 3^x \cdot \ln(3) \).

Step 1: Understand the function

The function in the exercise is \( y = 3^x \), here y is the dependent variable and x is independent. The task is to find derivative of y with respect to x.

Step 2: Write down the rule for derivative of an exponential function

The rule for finding the derivative of an exponential function is \( \frac{d}{dx}(a^x) = a^x \cdot \ln(a) \), where a is a constant, \( \ln \) is natural logarithm.

Step 3: Apply the rule to the function

Applying the rule of differentiation to this function, we have the derivative as \( \dot{y} = 3^x \cdot \ln(3) \).

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions where the base is a constant number, and the exponent is a variable, such as in the function \(y = a^x\), where \(a\) is the base, a constant, and \(x\) is the variable exponent. These functions are important in various fields because they describe growth or decay processes, such as population growth, interest rates, or radioactive decay, where the rate of change is proportional to the current amount.

Key characteristics of exponential functions include:

• They never touch the x-axis; their graphs approach but only get closer and closer to the x-axis.
• They grow (or decay) faster over time, representing processes with constant relative growth or decay rates.
• The base \(a\) determines the direction of the function, where \(a>1\) indicates growth and \(0

Understanding exponential functions paves the way to grasping more complex mathematical concepts associated with growth and change.

Natural Logarithm

The natural logarithm is a logarithm to the base \(e\), where \(e\) (approximately 2.718) is a mathematical constant. This logarithm is time-efficient and natural when dealing with exponential growth and decay, making it crucial in calculus for simplifying derivative calculations.

The natural logarithm, indicated as \(\ln(x)\), has fundamental properties that can be useful:

• \(\ln(e) = 1\), as you would expect because \(e^1 = e\).
• \(\ln(1) = 0\), because \(e^0 = 1\).
• The function is undefined for \(x \leq 0\), since logarithms of non-positive numbers aren't real.

Natural logarithms greatly simplify differentiation of exponential functions, given their neat properties that align well with the derivative processes, especially when finding the slopes of exponential growth curves.

Differentiation Rules

Differentiation rules are sets of guidelines used in calculus to find a derivative, which measures how a function changes as its input changes. Mastering these rules is critical for solving complex calculus problems.

For exponential functions like \(y = a^x\), the differentiation rule states that the derivative \(\frac{d}{dx}(a^x)\) is \(a^x \cdot \ln(a)\). This rule arises because the rate of change of exponential functions is proportional to their value, and the multiplicative effect of exponential growth is captured by the natural log.

It's also important to review basic differentiation rules:

• Power Rule: \(\frac{d}{dx}(x^n) = nx^{n-1}\) for any real number \(n\).
• Product Rule: \(\frac{d}{dx}(u \cdot v) = u'v + uv'\) for functions \(u\) and \(v\).
• Chain Rule: \(\frac{d}{dx}(f(g(x))) = f'(g(x)) \cdot g'(x)\), helpful for composite functions.

These rules, along with the rule for exponential functions, create a toolbox for discovering rates of change and understanding how different types of functions behave.