Problem 9 - Calculus

Question

Find the derivatives of the functions $$ 1 /\left(1+e^{x}\right) $$

Step-by-Step Solution

Answer

The derivative is \( f'(x) = \frac{-e^x}{(1+e^x)^2} \).

Step 1: Identify the Function

The given function is \( f(x) = \frac{1}{1+e^x} \). To differentiate this function, we will use the quotient rule, which is used for functions presented as a division.

Step 2: Apply the Quotient Rule

The quotient rule states that if you have a function \( u/v \), its derivative is \( \frac{v \cdot u' - u \cdot v'}{v^2} \). Here, \( u = 1 \) and \( v = 1+e^x \).

Step 3: Differentiate the Numerator and Denominator

Calculate the derivative of the numerator and the denominator: \(u' = 0\) (derivative of 1 is 0), and \(v' = \frac{d}{dx}(1 + e^x) = e^x\).

Step 4: Substitute and Simplify

Substitute into the quotient rule: \(\frac{(1+e^x) \cdot 0 - 1 \cdot e^x}{(1+e^x)^2} = \frac{-e^x}{(1+e^x)^2} \).Simplify further to get: \( f'(x) = \frac{-e^x}{(1+e^x)^2} \).

Key Concepts

Quotient Rule

The quotient rule is a technique in calculus used to find the derivative of a function that is the ratio of two other functions. When you have a function that can be expressed as \( \frac{u}{v} \), the quotient rule is essential. Understanding this rule is crucial when differentiating fractions, especially when the numerator and denominator are functions of \( x \).
Here’s how you apply the quotient rule:

• Identify the numerator (\( u \)) and the denominator (\( v \)) of the function.
• Find the derivatives \( u' \) and \( v' \). These are the derivatives of \( u \) and \( v \) with respect to \( x \).
• Substitute these into the quotient rule formula:

The formula for the quotient rule is:\[\left( \frac{u}{v} \right)' = \frac{v \cdot u' - u \cdot v'}{v^2}\]By following these steps, you can differentiate complex rational functions. In our exercise, we applied this rule to find the derivative of \( \frac{1}{1+e^x} \), starting by identifying \( u = 1 \) and \( v = 1+e^x \).

Exponential Function

An exponential function is any function of the form \( e^x \), where \( e \) is the base of the natural logarithm, approximately equal to 2.71828. These functions are unique because their rate of growth is proportional to their value.
Understanding exponential functions is key in calculus because they frequently appear in applications. Differentiating an exponential function is straightforward because the derivative of \( e^x \) with respect to \( x \) is simply \( e^x \).
This property makes exponential functions easy to work with when applying various differentiation rules, such as the quotient rule. For the function \( \frac{1}{1+e^x} \), recognizing the role of \( e^x \) was essential for accurately finding the derivative of the denominator \((v' = e^x)\). This simplicity helps streamline the differentiation process when exponential functions are involved.

Differentiation

Differentiation is the process of finding the derivative of a function. The derivative represents the rate of change of the function concerning its variable. In calculus, differentiation is a fundamental concept used to analyze how functions change and to tackle various problems involving rates.
There are many rules within differentiation, such as the product rule, chain rule, and quotient rule, each designed for specific types of functions. For example, the quotient rule, as seen in our problem, helps differentiate ratios of functions.
When beginning with differentiation, always start by identifying the components of the function that need to be differentiated. After identifying, apply the appropriate rule. In the exercise, we differentiated the function \( \frac{1}{1+e^x} \) by determining the components \( u \) and \( v \), finding their derivatives, and applying the quotient rule. This allowed us to find \( f'(x) = \frac{-e^x}{(1+e^x)^2} \). Differentiation enables us to understand and measure how functions behave dynamically.