91Ó°ÊÓ

Differentiation is a fundamental concept in calculus that involves the computation of a derivative. A derivative represents the rate at which a function changes at any given point. In simpler terms, it tells us the "slope" of the function at a specific location.
For the given function, where we have \( F(t) = \ln t \), differentiation helps us find the function \( f(t) \), which is the derivative of \( F(t) \). This process involves applying standard differentiation rules. The derivative of a logarithmic function \( \ln t \) is \( \frac{1}{t} \).

• Finding a derivative often involves rules like the power rule, chain rule, product rule, and quotient rule.
• In this exercise, we made use of the derivative of the natural logarithm.

With this knowledge, we concluded that \( f(t) = \frac{1}{t} \) is the derived function used for further calculations.

An antiderivative is essentially the reverse of differentiation. It is often referred to as integration, which is a core part of calculating areas under curves.
An antiderivative of a function \( f(t) \) is another function \( F(t) \) such that \( F'(t) = f(t) \). This means that if we differentiate \( F(t) \), we get back \( f(t) \).
In the given exercise, the function \( F(t) = \ln t \) is actually the antiderivative of \( f(t) = \frac{1}{t} \). When applying the Fundamental Theorem of Calculus, we need the antiderivative to evaluate the definite integral \( \int_{a}^{b} f(t) \, dt \).

• The Fundamental Theorem of Calculus links differentiation and integration, stating that if you have an antiderivative, you can calculate the definite integral of a function.
• It assures us that given \( f(t) \) continuous on an interval, then \( \int_{a}^{b} f(t) \, dt = F(b) - F(a) \).

This theorem was crucial in evaluating \( \int_{1}^{5} \frac{1}{t} \, dt \) in the problem by computing \( \ln 5 - \ln 1 \).

The logarithmic function \( \ln t \) represents the natural logarithm. Logarithms are mathematical functions that are the inverses of exponentiation. The base of the natural logarithm is \( e \), an irrational constant approximately equal to \( 2.718 \).
Logarithms answer the question, "To what exponent must the base \( e \) be raised to produce this number?" For instance, \( \ln 5 \) tells us the power to which \( e \) must be raised to equal 5.

• Logarithmic functions efficiently model growth and decay processes, making them useful in various scientific applications.
• \( \ln 1 = 0 \) because \( e^0 = 1 \), which simplifies calculations during integration as seen in our step-by-step solution.

The logarithmic properties, such as \( \ln(ab) = \ln a + \ln b \) and \( \ln(a/b) = \ln a - \ln b \), are crucial in simplifying expressions when calculating definite integrals. These properties were directly applied in simplifying our final answer to \( \ln 5 \) by recognizing that \( \ln 1 = 0 \).