In Exercises 65 and 66, find the derivative using the definition.


ƒ(t) = 1 .
2t + 1

Verified step by step guidance

1. Start by recalling the definition of the derivative. The derivative of a function ƒ(t) at a point t is given by the limit: lim(h→0) [(ƒ(t + h) - ƒ(t)) / h].
2. Substitute the given function ƒ(t) = 1/(2t + 1) into the definition. You need to find ƒ(t + h) first, which is 1/(2(t + h) + 1).
3. Calculate the difference ƒ(t + h) - ƒ(t). This will be: [1/(2(t + h) + 1)] - [1/(2t + 1)].
4. Simplify the expression for ƒ(t + h) - ƒ(t) by finding a common denominator, which is (2(t + h) + 1)(2t + 1).
5. Substitute the simplified expression into the derivative definition and evaluate the limit as h approaches 0: lim(h→0) {[(2t + 1) - (2(t + h) + 1)] / [h(2(t + h) + 1)(2t + 1)]}.

Key Concepts

Derivative Definition

The derivative of a function at a point is defined as the limit of the average rate of change of the function as the interval approaches zero. Mathematically, it is expressed as f'(a) = lim (h -> 0) [f(a + h) - f(a)] / h. This definition is fundamental for understanding how functions change and is the basis for calculating derivatives.

Limit

A limit is a fundamental concept in calculus that describes the behavior of a function as it approaches a particular point. It is essential for defining derivatives, as the derivative itself is a limit. Understanding limits helps in analyzing the continuity and behavior of functions near specific values.

Function Composition

Function composition involves combining two functions where the output of one function becomes the input of another. In the context of finding derivatives, it is important to understand how to manipulate functions, especially when applying the definition of the derivative to more complex expressions. This concept is crucial for simplifying the calculations involved in differentiation.