Problem 27 - Calculus and its applications

Question

Differentiate. $$ F(x)=4-e^{2 x} $$

Step-by-Step Solution

Answer

The derivative of \( F(x) = 4 - e^{2x} \) is \( F'(x) = -2e^{2x} \).

Step 1: Identify Terms in the Function

The function to differentiate is \( F(x) = 4 - e^{2x} \). Notice that this function consists of two terms: a constant term 4 and an exponential term \( -e^{2x} \). Differentiation will be applied separately to each term.

Step 2: Differentiate the Constant Term

The derivative of a constant is zero. Therefore, the derivative of the constant term 4 is 0. We can write it as: \( \frac{d}{dx}[4] = 0 \).

Step 3: Differentiate the Exponential Term

The second term is \( -e^{2x} \). The derivative of \( e^{2x} \) with respect to \( x \) is obtained by using the chain rule: the derivative of \( e^{u} \) with respect to \( u \) is \( e^{u} \), and the derivative of \( 2x \) with respect to \( x \) is 2. Thus, \( \frac{d}{dx}[e^{2x}] = 2e^{2x} \). The negative sign from the original term is carried through, giving: \( \frac{d}{dx}[-e^{2x}] = -2e^{2x} \).

Step 4: Combine the Derivatives

Add the derivatives of each term to find the derivative of the whole function. From Step 2, the derivative of the constant term is 0, and from Step 3, the derivative of the exponential term is \( -2e^{2x} \). Thus, the derivative of \( F(x) \) is \( F'(x) = 0 - 2e^{2x} \), simplifying to \( F'(x) = -2e^{2x} \).

Key Concepts

Chain Rule

In calculus, the chain rule is a fundamental method used for differentiating composite functions, which are functions composed of two or more functions. If you have a function in the form of a composition, such as \( f(g(x)) \), the chain rule helps you find the derivative by breaking it down into its individual parts.

• To differentiate \( f(g(x)) \), you first differentiate the outer function \( f \) with respect to \( g(x) \), treating \( g(x) \) as a single variable.
• Then, you multiply that derivative by the derivative of the inner function \( g(x) \) with respect to \( x \).

Let's apply this to the exponential term \( e^{2x} \) in the original exercise.

• Here, treat \( 2x \) as \( u \) in the function \( e^u \). The derivative of \( e^u \) with respect to \( u \) is just \( e^u \).
• Next, differentiate \( 2x \) with respect to \( x \), which gives 2.

Therefore, using the chain rule, the derivative of \( e^{2x} \) is simply \( 2e^{2x} \). It allows for differentiating functions with complex layers, making calculus a powerful tool for analyzing functions.

Exponential Function

Exponential functions are a key concept in mathematics, and they have a special form: \( y = e^{x} \). Here "e" represents Euler's number, approximately 2.718, which is the base of natural logarithms.

• Exponential functions are characterized by their constant rate of growth or decay.
• In calculus, they serve as a fundamental building block due to their ubiquity in natural processes and mathematical modeling.

The exponential function \( e^{x} \) is particularly notable because the rate of change of \( e^{x} \) is equal to itself and its derivative is also \( e^{x} \).

• This property makes exponential functions highly predictable and important, especially in fields such as finance, natural sciences, and engineering.

When differentiating functions that include exponential terms, such as \( e^{2x} \), it's crucial to remember that the chain rule will often be employed to handle the exponentiation. As seen in the original problem, this allows us to correctly find the rate of change for such functions.

Derivative of Constant

In calculus, the derivative of a constant function is zero. Constant functions are those where the output value does not change, regardless of the input.

• A simple example is \( f(x) = 4 \).
• No matter what value \( x \) takes, the output remains 4.

In terms of differentiation, this means there is no change and hence the rate of change, or the derivative, is zero.

• For any constant \( C \), \( \frac{d}{dx}[C] = 0 \).

This concept is straightforward and essential because it simplifies differentiation problems. In the given exercise, differentiating the constant term 4 correctly resulted in zero, simplifying the computation of the entire function’s derivative.