Find the derivative of the following functions by first expanding or simplifying the expression. Simplify your answers. 
r(t) = (e^2t + 3e^t + 2) / (e^t + 2)

Verified step by step guidance

1. Step 1: Identify the function r(t) = \frac{e^{2t} + 3e^{t} + 2}{e^{t} + 2}. This is a rational function, which can be simplified by performing polynomial long division if possible.
2. Step 2: Perform polynomial long division on the numerator (e^{2t} + 3e^{t} + 2) by the denominator (e^{t} + 2). Divide the leading term of the numerator by the leading term of the denominator to find the first term of the quotient.
3. Step 3: Multiply the entire divisor (e^{t} + 2) by the first term of the quotient obtained in Step 2, and subtract this product from the original numerator to find the new remainder.
4. Step 4: Repeat the division process with the new remainder obtained in Step 3, continuing until the degree of the remainder is less than the degree of the divisor.
5. Step 5: Once the expression is simplified, differentiate the resulting expression term by term using basic differentiation rules, such as the power rule and the constant rule, to find the derivative r'(t).

Key Concepts

Derivative

The derivative of a function measures how the function's output value changes as its input value changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. In calculus, derivatives are fundamental for understanding rates of change and are used in various applications, including optimization and motion analysis.

Simplification of Expressions

Simplifying expressions involves rewriting them in a more manageable or understandable form, often by combining like terms, factoring, or using algebraic identities. In the context of derivatives, simplification can make it easier to apply differentiation rules and can lead to more straightforward calculations. This step is crucial for ensuring accuracy and efficiency in finding derivatives.

Quotient Rule

The quotient rule is a formula used to find the derivative of a function that is the ratio of two other functions. If a function is expressed as f(x) = g(x)/h(x), the derivative is given by f'(x) = (g'(x)h(x) - g(x)h'(x)) / (h(x))^2. This rule is essential when differentiating functions that are expressed as fractions, ensuring that both the numerator and denominator are correctly accounted for in the differentiation process.