Problem 11 - Thomas Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify the Form

Recognize that the function given is of the form \(y = a^x\), specifically \(y = 2^x\). This is an exponential function where the base \(a\) is a constant and the exponent is the variable.

Step 2: Recall the Derivative Formula for Exponential Functions

The derivative of an exponential function \(a^x\) with respect to \(x\) is given by \(\frac{d}{dx}[a^x] = a^x \ln(a)\). Here, \(a\) is the base of the exponential function and \(x\) is the exponent.

Step 3: Substitute the Values

Substitute the values from the function into the formula. Given that \(y = 2^x\), it follows that the derivative \(\frac{dy}{dx} = 2^x \ln(2)\).

Step 4: Write the Final Result

The derivative of \(y = 2^x\) with respect to \(x\) is \(\frac{dy}{dx} = 2^x \ln(2)\).

Key Concepts

Exponential Function

An exponential function is a mathematical function of the form \( y = a^x \), where \( a \) is a positive constant, known as the base of the exponential, and \( x \) is the exponent, which is typically a variable. Unlike polynomial functions, where the variable is in the base, exponential functions have the variable in the exponent.Exponential functions exhibit constant percentage growth or decay. This means that as \( x \) increases or decreases, the value of the function changes rapidly. Because of this property, exponential functions are widely used in natural sciences to model growth processes, such as population growth, radioactive decay, and compound interest in finance.Some key characteristics of exponential functions include:

• Exponential Growth: When \( a > 1 \), the function model growth behavior.
• Exponential Decay: When \( 0 < a < 1 \), the function models decay behavior.
• Horizontal Asymptote: The x-axis (y=0) acts as a horizontal asymptote for the function as \( x \) approaches negative infinity.

Exponent Rules

Exponent rules are the set of guidelines that define how to manipulate expressions involving exponents. These rules are essential in simplifying complex expressions and solving equations involving exponential terms.Some fundamental exponent rules include:

• Product of Powers Rule: \( a^m \times a^n = a^{m+n} \). This rule states that when multiplying two powers with the same base, add the exponents.
• Power of a Power Rule: \( (a^m)^n = a^{m \cdot n} \). This implies that you multiply the exponents when raising a power to another power.
• Quotient of Powers Rule: \( \frac{a^m}{a^n} = a^{m-n} \), given that \( a eq 0 \). This rule applies when dividing two powers with the same base, where you subtract the exponents.
• Zero Exponent Rule: \( a^0 = 1 \) for any non-zero \( a \). This means any number raised to the power of zero equals one.

These rules help simplify the process of deriving exponential functions, particularly when managing the exponent terms separately before applying derivative formulas.

Derivative Formula

The derivative formula for exponential functions is a vital tool in calculus that helps determine the rate of change of these functions with respect to their variables. For an exponential function \( y = a^x \), the derivative, or \( \frac{dy}{dx} \), is calculated using the formula:\[\frac{d}{dx}[a^x] = a^x \ln(a)\]In this formula, \( a \) represents the base of the exponential function, while \( \ln(a) \) is the natural logarithm of \( a \). It's significant to note that the derivative of \( a^x \) involves both the original function itself, \( a^x \), and the constant \( \ln(a) \).Understanding this concept is crucial because:

• If \( a > 1 \), the natural logarithm \( \ln(a) \) is positive, indicating positive growth.
• If \( 0 < a < 1 \), \( \ln(a) \) is negative, indicating decay as the function's rate of change decreases.

When specifically dealing with a function like \( y = 2^x \), the derivative becomes \( 2^x \ln(2) \), providing the formula needed for assessing its rate of change.