Post 1: Introduction to Differentiation

Introduction: Differentiation is a fundamental concept in calculus that allows us to analyze the rate of change of a function. It provides a tool for understanding how a function behaves at various points and is crucial in many fields such as physics, economics, and engineering. In this post, we will explore the definition of the derivative and how it represents the rate of change of a function.

Derivative Definition: The derivative of a function represents its instantaneous rate of change at any given point. It measures how the function's output changes as its input varies. Mathematically, the derivative of a function f(x) is denoted as f'(x) or dy/dx (read as "dy by dx"). The derivative is found by taking the limit as the change in x approaches zero:

f'(x) = lim(h->0) [f(x + h) - f(x)] / h

This formula calculates the slope of a tangent line to the graph of the function at a particular point.

Rate of Change Interpretation: The derivative represents the rate at which a function is changing with respect to its input variable. For example, suppose we have a function that represents the position of an object over time. The derivative of this function would give us the object's velocity, which tells us how fast the object is changing its position at each moment.

Example: Let's consider the function f(x) = 3x^2. We want to find its derivative, f'(x), using the definition of the derivative:

f'(x) = lim(h->0) [(3(x + h)^2 - 3x^2) / h]

Expanding and simplifying this expression, we get:

f'(x) = lim(h->0) [3(x^2 + 2hx + h^2 - x^2) / h] = lim(h->0) [6hx + 3h^2 / h] = lim(h->0) 6x + 3h = 6x

Hence, the derivative of f(x) = 3x^2 is f'(x) = 6x. This means that at any given point on the graph of f(x), the rate of change is given by 6 times the value of x.

Conclusion: Differentiation is a powerful tool in calculus that allows us to understand the rate of change of a function. The derivative represents this rate of change and is essential in analyzing various phenomena in mathematics and real-world applications. In the next posts, we will explore different derivative rules and their applications. Stay tuned!