Question

Compute the derivative of the given function. $$f(r)=6 e^{r}$$

Short answer

The derivative of \( f(r) = 6e^r \) is \( 6e^r \).

Step-by-step solution

Recall the derivative rule for exponential functions

The derivative of an exponential function with base \( e \) is the exponential function itself. Formally, if \( f(x) = e^x \), then \( f'(x) = e^x \). Additionally, if the function has a constant multiplier, the derivative of \( af(x) \) is \( a \cdot f'(x) \).

Identify the components of the function

The function given is \( f(r) = 6e^r \), where 6 is a constant multiplier and \( e^r \) is the exponential term.

Apply the differentiation rule

Using the rule from Step 1, the derivative of \( e^r \) is \( e^r \). Therefore, the derivative of \( 6e^r \) is \( 6 \cdot e^r \), keeping the constant multiplier 6 and the base exponential function unchanged.

Key concepts

Exponential Functions

Exponential functions are mathematical operations where a constant base is raised to a variable exponent. The most common base is Euler's number, denoted as \( e \), which is approximately 2.718. This base is used frequently because of its unique property where its derivative (slope of the tangent at any point on the curve) and integral (area under the curve) are equivalent to the function itself.
Such functions are often written in the form of \( e^x \) or in our case \( e^r \), where the exponent is the variable. Exponential functions grow rapidly, making them useful in modeling many real-world phenomena like population growth or radioactive decay.
When dealing with financial formulas, biology, and complex systems, recognizing and handling exponential functions is crucial. The characteristic that \( e^x \)'s derivative stays \( e^x \) is particularly important for calculus as it simplifies the differentiation process.

Derivative Rules

Derivative rules, often called differentiation rules, are guidelines for finding the derivative of a function. Derivatives represent the rate of change or the slope of a function at a particular point.
Basic derivative rules include:

• The power rule: For \( f(x) = x^n \), the derivative \( f'(x) = nx^{n-1} \).
• The constant rule: The derivative of a constant is zero, since its rate of change is zero.
• The exponential rule, especially for natural exponentials, as seen in \( e^x \): The derivative is simply \( e^x \) because it grows at a rate proportional to its value.

By using these rules, we can break down complex expressions into manageable parts, calculating derivatives swiftly and accurately. Understanding and applying these rules is key to mastering calculus, especially when analyzing exponential functions.

Constant Multiplier in Differentiation

The constant multiplier rule in differentiation is a simple yet powerful concept. It states that when a function is multiplied by a constant, the constant can be factored out when taking the derivative. This rule helps us navigate functions like \( f(x) = af(x) \), where \( a \) is a constant.
For example, in the function \( f(r) = 6e^r \), the constant 6 remains unaffected in the differentiation process. According to the constant multiplier rule, the derivative is found by multiplying 6 by the derivative of \( e^r \), which remains \( e^r \). Thus, the derivative is \( 6e^r \).
This rule simplifies the calculation of derivatives by allowing you to focus on the variable part of the function first. It streamlines the differentiation of functions with constant factors, making calculus more efficient and accessible.