Question

Derivatives Find and simplify the derivative of the following functions. $$g(x)=\frac{e^{2 x}-1}{e^{x}-1}$$

Short answer

Question: Find the simplified derivative of the function $$g(x) = \frac{e^{2x} - 1}{e^x - 1}$$. Answer: The simplified derivative of the function $$g(x)$$ is $$ g'(x) = \frac{e^{3x} - e^{2x}}{(e^x - 1)^2} $$.

Step-by-step solution

Find the derivatives of f(x) and h(x)

To compute the derivatives of $$f(x)$$ and $$h(x)$$, we use the basic properties of derivatives. Recall that the derivative of $$e^x$$ is again $$e^x$$, and the derivative of a constant is always 0. $$f'(x) = \frac{d}{dx}(e^{2x} - 1) = 2e^{2x}$$ $$h'(x) = \frac{d}{dx}(e^{x} - 1) = e^{x}$$

Apply the quotient rule

Now that we have both $$f'(x)$$ and $$h'(x)$$, we can apply the quotient rule to find the derivative $$g'(x)$$: $$g'(x) = \frac{f'(x)h(x) - f(x)h'(x)}{(h(x))^2} = \frac{(2e^{2x})(e^x - 1) - (e^{2x} - 1)(e^x)}{(e^x - 1)^2}$$

Simplify the derivative

Next, we simplify the expression for the derivative, $$g'(x)$$: First, distribute the terms in the numerator: $$g'(x) = \frac{2e^{3x} - 2e^{2x} - e^{3x} + e^{2x}}{(e^x - 1)^2}$$ Now, combine like terms: $$g'(x) = \frac{e^{3x} - e^{2x}}{(e^x - 1)^2}$$ So, the simplified derivative of the function $$g(x)$$ is: $$ g'(x) = \frac{e^{3x} - e^{2x}}{(e^x - 1)^2} $$

Key concepts

Derivatives

Derivatives are a fundamental concept in calculus that measure how a function changes as its input changes. They tell us the rate at which the value of a function is changing at any given point. In the context of this exercise, we are asked to find the derivative of a function that is given as a quotient of two other functions.To do this, we first need to understand the basic properties of derivatives. For example, the derivative of an exponential function, such as \( e^x \), is straightforward because the derivative of \( e^x \) is itself \( e^x \). This simplicity makes exponential functions particularly handy in calculus.When dealing with quotients of functions, it's useful to apply the **quotient rule**. If \( g(x) = \frac{f(x)}{h(x)} \), then the derivative \( g'(x) \) is given by:\[g'(x) = \frac{f'(x)h(x) - f(x)h'(x)}{(h(x))^2}.\]Applying these principles helps us find the derivative of complex expressions in a structured way, as shown in the solution.

Exponential Functions

Exponential functions are a type of function where a constant base is raised to a variable exponent. The most famous exponential function is \( e^x \), where \( e \) is Euler's number, approximately 2.718, a mathematical constant that appears in various natural processes.The function \( e^x \) is unique because it is its own derivative. This property makes calculations involving exponential functions easier and is useful in a wide range of mathematical problems, such as compounding interest, population growth models, and in this case, calculus problems.In the given problem, both the numerator and the denominator of the function \( g(x) \) involve exponential terms \( e^{2x} \) and \( e^x \). Calculating derivatives of these terms, as shown, involves simply multiplying by the derivative of the exponent due to the chain rule. For \( e^{2x} \), the derivative is \( 2e^{2x} \), found by multiplying \( e^{2x} \) by the derivative of \( 2x \), which is 2.

Simplifying Expressions

Simplifying expressions, especially after taking derivatives, is crucial to make the results more interpretable and manageable. It often involves combining like terms and performing algebraic manipulations to present the derivative in its simplest form.In the exercise, after applying the quotient rule, the resulting expression for the derivative \( g'(x) \) needs to be simplified:\[g'(x) = \frac{2e^{3x} - 2e^{2x} - e^{3x} + e^{2x}}{(e^x - 1)^2}.\]The simplification involves:

• Distributing terms in the numerator to organize like terms.
• Combining similar exponential terms, allowing \(2e^{3x} - e^{3x}\) to become \(e^{3x}\), and \(-2e^{2x} + e^{2x}\) simplifying to \(-e^{2x}\).

These steps result in the cleaned-up expression:\[g'(x) = \frac{e^{3x} - e^{2x}}{(e^x - 1)^2},\]clearly showcasing the process of simplification and making the expression easier to interpret.