Problem 15 - Calculus and its applications

Question

Differentiate. $$ f(x)=6 e^{x} $$

Step-by-Step Solution

Answer

The derivative is \(f'(x) = 6e^x\).

Step 1: Identify the Function to Differentiate

We are given the function \(f(x) = 6e^x\). Our goal is to find its derivative \(f'(x)\).

Step 2: Recall the Derivative Rule for Exponential Functions

The derivative of an exponential function \(e^x\) with respect to \(x\) is \(e^x\) itself. That is, \(\frac{d}{dx} e^x = e^x\).

Step 3: Apply the Constant Multiple Rule

According to the differentiation rules, when differentiating a constant multiplied by a function, you can multiply the constant by the derivative of the function. Therefore, for \(f(x) = 6e^x\), you apply this rule to get the derivative: \(f'(x) = 6 \cdot \frac{d}{dx}(e^x)\).

Step 4: Differentiate \(e^x\) and Simplify

Differentiate \(e^x\) to get \(e^x\) and substitute back into the equation: \(f'(x) = 6 \cdot e^x\).

Step 5: Write the Final Answer

The derivative of the function \(f(x) = 6e^x\) is \(f'(x) = 6e^x\).

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. One of the most common and frequently encountered exponential functions in calculus is the function involving the base of the natural logarithm, known as Euler's number, denoted as \( e \). The function \( e^x \) is unique because the rate at which it changes is directly proportional to its current value. This means that its derivative is the same as the original function.

• This property makes it particularly simple to work with when differentiating.
• Exponential functions like \( e^x \) grow very quickly as \( x \) increases.
• They are commonly used in populations growth models, financial mathematics, and more.

When faced with an exponential function involving \( e \), such as \( f(x) = 6e^x \), the derivative follows a straightforward process due to its intrinsic properties.

Derivative Rules

Derivative rules are essential principles in calculus that allow us to find the rate of change or the slope of the function at any point. They provide a systematic way to differentiate functions without having to calculate limits every time. One of the most important rules is the derivative of the exponential function. When differentiating \( e^x \), the key takeaway is:

• The derivative of \( e^x \) is simply \( e^x \).
• This property remains regardless of the coefficient in front of \( e^x \), due to the constant multiple rule.

Coupled with other derivative rules like the power rule, product rule, and quotient rule, mastering derivative rules simplifies the process of differentiation when tackling a variety of complex functions.

Constant Multiple Rule

The Constant Multiple Rule is a fundamental differentiation rule used to simplify the process of finding derivatives when a constant is involved. This rule states that when differentiating a function that is multiplied by a constant, you can take the constant outside of the differentiation operation. Mathematically, if \( c \) is a constant and \( u(x) \) is a function, then:

• \( \frac{d}{dx} [c \cdot u(x)] = c \cdot \frac{d}{dx} [u(x)] \)

In the example of \( f(x) = 6e^x \), applying the constant multiple rule allows us to treat the constant 6 separately:

• First, differentiate \( e^x \) to get \( e^x \).
• Then multiply the result by 6 to get \( 6e^x \).

This rule greatly streamlines the differentiation process, especially when dealing with more complex functions where constants are involved. Recognizing and applying the Constant Multiple Rule helps avoid common mistakes and simplifies calculus operations.