微积分是研究连续变化的数学分支，由牛顿和莱布尼茨在17世纪独立发展。它包含两个基本分支：微分学，研究变化率，如运动汽车的速度；积分学，处理量的累积，如求曲线下的面积。
极限是微积分的基础概念。它描述当自变量趋近某个值时，函数值的趋向。例如，当x趋近于1时，x平方的极限是1。即使函数在该点没有定义，我们仍可以研究其极限行为。
导数表示函数在某点的瞬时变化率。几何上，导数就是该点处切线的斜率。例如，x平方的导数是2x，在x等于2处，斜率为4。导数帮助我们理解函数的局部行为。
积分用于计算曲线下的面积，是微分的逆运算。我们可以用矩形来近似这个面积，矩形越多，近似越精确。定积分给出精确的面积值。积分在物理学中表示累积量，如位移、功等。
微积分在现代科学技术中有广泛应用。在物理学中用于描述运动和力学；在工程学中用于优化设计，如找到函数的最大值或最小值；在经济学中进行边际分析；在生物学中建立增长模型。微积分是理解变化世界的强大工具。
Limits are the cornerstone of calculus. They describe what happens to a function as the input approaches a specific value. The formal definition states that the limit of f(x) as x approaches a equals L if we can make f(x) arbitrarily close to L by choosing x sufficiently close to a. This concept allows us to analyze function behavior even at points where the function might not be defined, forming the foundation for derivatives and integrals.
Derivatives are defined as the limit of difference quotients. As we make the interval h smaller and smaller, the secant line approaches the tangent line. The derivative f prime of x equals the limit as h approaches zero of f of x plus h minus f of x, all divided by h. This gives us the instantaneous rate of change at any point. For example, the derivative of x squared is 2x, so at x equals 2, the slope is 4.
Derivative rules provide efficient methods for computing derivatives. The power rule states that the derivative of x to the n is n times x to the n minus 1. The product rule handles products of functions: the derivative of u times v equals u prime times v plus u times v prime. The chain rule deals with composite functions: the derivative of f of g of x equals f prime of g of x times g prime of x. The quotient rule applies to fractions: the derivative of u over v equals u prime v minus u v prime, all over v squared. These rules allow us to differentiate complex expressions systematically.
Derivatives have powerful real-world applications. In optimization problems, we find maximum and minimum values by setting the first derivative equal to zero to locate critical points. The second derivative test confirms whether these points are maxima or minima. Related rates problems involve finding how one quantity changes with respect to another, like the rate at which a balloon's volume increases as air is pumped in. Derivatives also help analyze curve behavior, determining where functions are increasing or decreasing, and their concavity. In physics, derivatives connect position, velocity, and acceleration, providing insights into motion and forces.