【其他】微积分

"So far as the theories of mathematics are bout reality, they are not certain: so far as they are certain, they are not about reality."

导数

汽车运行问题

$$\frac{ds}{dt}(t)=\frac{s(t+dt)-s(t)}{dt}$$

$dt$：一个有限小的非零量（区分无穷小）

图像求导问题

$f(x)=x^2$ 的导数是什么？

$f(x)=x^3$ 的导数是什么？

$$\frac{d(x^n)}{dx}=nx^{n-1}$$

$$(x+dx)^n=\sum_{i=0}^{n}C_n^i(x)^i(dx)^{n-i}$$

$f(x)=x^{-1}$ 的导数是什么？

链式法则和乘积法则

"Using the chain rule is like peeling an onion: you have to deal with each layer at a time, and if it is too big you will start crying."

$$\frac{df}{dx}=\frac{dg}{dx}+\frac{dh}{dx}$$

$$\frac{df}{dx}=g(x)\frac{dh}{dx}+h(x)\frac{dg}{dx}$$

$$\frac{df}{dx}(x)=\frac{dg}{dh}(h(x))\frac{dh}{dx}(x)$$

指数函数求导

"Who has not been amazed to learn that the function $y=e^x$, like a phoenix rising again from its won ashes, is its own derivative?"