Problem 16 - Calculus and its applications

Question

Differentiate. $$ f(x)=4 e^{x} $$

Step-by-Step Solution

Answer

The derivative of \( f(x) = 4e^x \) is \( 4e^x \).

Step 1: Identify the Function to Differentiate

The function given is \( f(x) = 4e^x \). Here, \( 4e^x \) is an exponential function, where \( e \) is the base of the natural logarithm.

Step 2: Differentiate the Exponential Function

To differentiate \( f(x) = 4e^x \), we will apply the rule for differentiating exponential functions. The derivative of \( e^x \) is \( e^x \). Thus, the derivative of \( 4e^x \) will be \( 4 \cdot e^x \).

Step 3: Write the Result

The differentiated function, or the derivative of \( f(x) = 4e^x \), is \( f'(x) = 4e^x \).

Key Concepts

Exponential Function

An exponential function is a mathematical function of the form

• \( f(x) = a \cdot e^{bx} \)

where \( e \) is a special constant approximately equal to 2.71828, known as Euler's number. The variable \( x \) is in the exponent position, which makes the function grow rapidly as \( x \) increases, or decrease quickly if \( b \) is negative. Exponential functions are widely used to model situations involving rapid growth or decay, such as population growth, radioactive decay, and interest compounding.
In our exercise, the function is \( f(x) = 4e^x \), where the constant 4 is a coefficient, and the base \( e \) results in a continuous growth. Recognizing the base and exponent is key to differentiating these types of functions smoothly.

Derivative

Derivatives are a fundamental tool in calculus used to determine the rate at which a function is changing at any given point. For any function \( f(x) \), its derivative \( f'(x) \) is another function that gives the slope of the tangent line to the curve of \( f(x) \) at each point.

• They provide vital insight into the behavior of functions such as increasing or decreasing trends.
• They are also used to find local maxima and minima, which are crucial for optimization problems.

The derivation rule for exponential functions specifically states that if you have an expression of the form \( e^{g(x)} \), then the derivative is \( e^{g(x)} \cdot g'(x) \). In simple terms, you "take down" the exponent, keep it as the coefficient, and include any derivative of the exponent in the final result.
Thus, differentiating our given function \( f(x) = 4e^x \) using this rule, the derivative is straightforward, resulting in \( f'(x) = 4e^x \), emphasizing the simplicity and elegance of working with the natural base \( e \).

Natural Logarithm

The natural logarithm, denoted as \( \ln(x) \), is the logarithm to the base \( e \). It is the inverse operation to exponentiation involving \( e \). Basic properties include:

• \( \ln(e) = 1 \) because \( e^1 = e \).
• \( \ln(1) = 0 \) because \( e^0 = 1 \).

Understanding natural logarithms is crucial when dealing with exponential growth and decay problems because of their close relationship with exponential functions. Specifically, they help convert products into sums, making complex algebra simpler to handle.
While our differentiation problem here didn't directly include natural logarithms, knowing the connection between \( e^x \) and \( \ln(x) \) strengthens comprehension of why the differentiation of \( e^x \) remains \( e^x \). With the natural base \( e \), exponential functions retain their form even when differentiated, showcasing the natural beauty and simplicity of these mathematical concepts.