Some knowledge about derivatives
- Remember a remote lecture
definition:
f ( x ) is a function whose derivative is denoted f ' ( x )
The derivative f ' ( x ) is the slope of the function f ( x ) in the function graph
The definition of the derivative function is : ( f( x + Δx ) - f( x ) ) / ( ( x + Δx ) - x )
Equivalent to taking two points in the function ( x1, y1), ( x2, y2 ), the formula is: ( y2 - y1 ) / ( x2 - x1 )
First, let's talk about limits:
lim x -> x0 f(x), indicating the limit of the function f(x) at the point x0.
(lim is just a notation, it is the abbreviation of the Latin limes, which almost means limit, but don't think it is limit, it makes people laugh)
The meaning of the expression is very simple, that is, what is the value of f(x) when the value of x is "infinitely close" to x0. For example f(x) = x is 0 when x->0.
Of course the limit may not exist. For example lim x->0 1 / x .
Everyone should know that when x infinitely tends to 0, the function value is infinite. Note that infinity is not necessarily the only case where the limit does not exist. There are also some cases such as the limit of the piecewise function at the discontinuous point does not exist. For example f(x) = x+1 if x > 0 = x - 1 if x < 0 x->0, because -1 is also true and 1 is also true.
This is actually called the left limit is not equal to the right limit, that is, when x approaches 0 from the left of the number axis and approaches from the right, the obtained function values are different.
Now let's talk about something new:
In order to express the concept of "small", we introduce a new symbol dx. We only need to understand him emotionally, and don't think about the mathematics behind it.
We can simply understand that dx is a very small value, such as lim x->1 f(x) = x, we can also write it as lim dx -> 0 f(1 + dx).
Then, let's do the question
Oh, by the way, when we write dx, the default dx is -> 0. Of course, this expression is not very precise, but we can easily write some mathematical expressions.
dx / dx = ？
dx / 1 = ？
When you see dx / 1, it automatically thinks, lim x-> 0 x / 1, so dx / dx is lim x->0 x / x, so dx / dx = 1, or they are both equal to 0. In fact, just follow lim x->0 x / x.
The next question is a bit tricky (dx)^2 / dx = ? , i.e. lim x->0 x^2/x = ?
Of course they are all equal to 0.
In fact, I want to illustrate a very important concept: infinitesimals have "orders". (dx)^2 is called 2nd-order infinitesimal, which means that it is a little smaller than dx, and when dx^2 / dx, there is only 0.
Next, we need to apply the definition of derivative and its definition formula we said at the beginning:
What is the derivative of f(x) = x^2?
At this time, a little friend wants to ask: "Do you need its slope? But I won't ask for the quadratic function 2333..."
This is where definitions are needed. Substitute f(x) = x^2 into the definition ( f(x + dx) - f(x)) / ((dx + x) - x) , the answer is dx + 2x , and because lim dx -> 0, So, the answer should be 2x. (pictured)
Geometrically speaking, this is probably what it means: when you get two points infinitely close, the slope is equal to 2x
OK, next question
What is the derivative of f(x) = cos(x)? ? (@_@)
再次请出定义式，so... f'(x) = - sin(x) 你有没有做对呢？？
PS：此处要补充一个知识：sin(dx) 近似于 dx
推导过程：
老师的字很漂酿 是吧是吧 (*￣︶￣)
接下来是思考题：
T：我们来思考一个问题 f'(x) = 0说明什么？？
M：说明 x 是常数
T: 如果只是某个点是0呢？？
M: 斜率是0
T: 说明啥 （老师块忍不了了 2333）
M: @_@ 凌乱ing
T: 好吧， 说明x是一个 f(x)的极值点，极大值/极小值。嗯...比如我们来求一下 f(x) = ax^2 + bx + c 的导数。
M: 2ax+b
T: f‘(x) = 0 的解是什么
M: - b / 2a （大家有没有发现什么呢）
T: 嗯。所以 求导数 可以用来求函数的极值，不过 极值和“最值”是不一样的哦。有一个定理，一个函数的最大/最小值 一定出现在 端点处 或者 极值处。应该很好理解吧
（ENDING：我觉得接下来的东西你基本 都能看懂了，记得如果有想不明白的就回到 定义上，想想是怎么算这些函数的导数的（就像是 不忘初心，方得始终）。我觉得你应该 网上的东西 都能看懂了。嗯 还有就是 极值点不一定是最值点，极值点可以用 f'(x) = 0 找到）