国际课程提前学微积分1: 导数定义（后附视频讲解）

今天的国际课程提前学，给大家带来的是：导数的定义，请大家一起来学习吧！（后附视频解读哦~）

微积分（Calculus）是数学中重要的主线，包括微分与积分两部分，微分的核心作用是找到

The gradient of a line直线的斜率

If (c, f(c)) is the point on the curve and is a second point on the graph of f, the gradient of the secant line through the two points is given by substitution into the gradient formula.

已知曲线上两点，连接成一条直线，则这条直线（线段）的斜率可以通过斜率公式求出。

The gradient of a curve  

曲线的斜率

we draw the straight line y = mx + c passing along the bottom of the log, then this line is a tangent to the curve at the point of contact. The gradient m of the tangent is the gradient of the curve at the point of contact.

一个木头管道，在底部做一条切线，这条切线的斜率就是这条曲线在这一点的斜率。

Example

Take the point P(3,9) and another point Q close to (3,9) on the curve. Let the x coordinate of Q be 3+h where h is small. Since y = x2 at Q，the y coordinate of Q will be (3+h)2 .

If h=0.001, the gradient of PQ is 6.001, and when h=-0.001, the gradient of PQ is 5.999. The gradient of the tangent at P is between these two values.

As approaches 0, Q approaches P, and secant line QP approaches tangent line. Then, the gradient of secant line QP approaches the gradient of tangent line ,Thus, the gradient of the tangent at (3,9) would be 6.

以二次函数y = x2为例，找曲线在点P（3.9）的斜率，在曲线上找另外一点Q，连接PQ，求出写段PQ的斜率表达式，当Q点无限接近P点，则线段PQ的斜率等于点P处的斜率，即h等于零时。

求If上任意一点的斜率，

The gradient function, or f’(x) is called the derivative of y with respect to x.

The process of finding the derivative of a function is called differentiation.

This “new” function gives the gradient of the tangent line to the graph of f at the point (x, f(x)), provided that the graph has a tangent line at this point.

斜率函数f’(x)叫做y关于x的导数，求导的过程叫做微分，这个新的函数是原函数上各个点的斜率组成的新函数。

这就是微积分中微分的核心作用，用来求曲线上的任意一点的斜率即瞬时变化率。