Problem 8 - Calculus

Question

Find the derivatives of the functions $$ 4^{4 x} $$

Step-by-Step Solution

Answer

The derivative is \(4^{4x} \cdot 4 \ln(4)\).

Step 1: Identify the Base and the Exponent

The given function is \( 4^{4x} \). Here, the base is \( 4 \) and the exponent is \( 4x \). This shows that our function is an exponential function where \( x \) appears in the exponent.

Step 2: Apply the Exponential Rule for Derivatives

For an exponential function \( a^{u(x)} \), the derivative is \( a^{u(x)} \cdot \ln(a) \cdot u'(x) \). In this case, \( a = 4 \) and \( u(x) = 4x \).

Step 3: Differentiate the Exponential Function

First, differentiate \( u(x) = 4x \). The derivative of \( 4x \) with respect to \( x \) is simply \( 4 \).

Step 4: Apply the Derivative Formula

Using the formula from Step 2, we find:\[ \frac{d}{dx}[4^{4x}] = 4^{4x} \cdot \ln(4) \cdot 4 \]Thus, the derivative of \( 4^{4x} \) is \( 4^{4x} \cdot 4 \ln(4) \).

Key Concepts

Exponential Function

An exponential function is a mathematical expression where a constant base is raised to a variable exponent. In many scientific and financial contexts, exponential functions are used to model growth or decay, such as population growth, radioactive decay, or compound interest.

The standard form of an exponential function is written as:

• General form: \(a^{x}\), where \(a\) is a positive constant
• Specific form for this example: \(4^{4x}\), where 4 is the base, and \(4x\) is the variable exponent

In our exercise, 4 serves as the base and \(4x\) is the exponent, showing that the function grows exponentially as \(x\) increases. Understanding exponential functions is crucial in calculus as they frequently appear in mathematical models. They are characterized by their unique property, where the rate of change is proportional to their current value.

Exponential Rule for Derivatives

The exponential rule for derivatives is a key tool for differentiating exponential functions. When you have a function of the form \(a^{u(x)}\), the derivative can be found using a specific formula. Let's break it down:

• The formula is: \(a^{u(x)} \cdot \ln(a) \cdot u'(x)\)
• Here, \(a\) is the base of the exponential function, and \(u(x)\) is the exponent which is a function of \(x\)
• \(\ln(a)\) stands for the natural logarithm of the base
• \(u'(x)\) represents the derivative of the exponent function

This rule is essential because it allows us to handle more complex exponential expressions beyond simple power functions. By applying this formula effectively, we find the rate of change for complex systems modelled by exponential functions.

Differentiation

Differentiation is a fundamental concept in calculus that involves finding the derivative of a function. A derivative represents the rate at which function values change with respect to changes in their variable. This process is pivotal in various fields such as physics, engineering, and economics, where predicting and analyzing change is crucial.

Key points about differentiation include:

• Derivatives tell us the slope of the function’s graph; it shows how fast or slow a function is changing at any point
• It involves different rules and techniques, such as the power rule, product rule, quotient rule, and chain rule
• For exponential functions like \(4^{4x}\), differentiation requires the use of the exponential rule

In our exercise, we differentiated \(u(x) = 4x\) to get \(4\). Understanding differentiation helps us lay the foundation for more advanced calculus topics, enabling students to tackle real-world problems involving rate of change.