# x^n 的导数

## x^2的导数

f^\prime(x)相当于\frac{\mathrm{d}}{\mathrm{d}x}f(x)

\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}f(x)=\frac{\mathrm{d}f(x)}{\mathrm{d}x}

\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}f(x)=\frac{\mathrm{d}}{\mathrm{d}x}x^2=\lim\limits_{\Delta x\to0}\frac{(x+\Delta x)^2-x^2}{\Delta x}\\{}\\ =\lim\limits_{\Delta x\to0}\frac{(x^2+2x\Delta x+(\Delta x)^2)-x^2}{\Delta x}\\{}\\ =\lim\limits_{\Delta x\to0}\frac{2x\Delta x+(\Delta x)^2}{\Delta x}\\{}\\ =\lim\limits_{\Delta x\to0}2x+\Delta x\\{}\\ =2x

\displaystyle (a+b)^3=(a+b)^2(a+b)\\{}\\ =a^3+2a^2b+ab^2+a^2b+2ab^2+b^3\\{}\\ =a^3+3a^2b+3ab^2+b^3

## x^3的导数

\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}x^3=\lim\limits_{\Delta x\to0}\frac{(x+\Delta x)^3-x^3}{\Delta x}\\{}\\ =\lim\limits_{\Delta x\to0}\frac{x^3+3x^2\Delta x+3x(\Delta x)^2+(\Delta x)^3-x^3}{\Delta x}\\{}\\ =\lim\limits_{\Delta x\to0}\frac{3x^2\Delta x+3x(\Delta x)^2+(\Delta x)^3}{\Delta x}\\{}\\ =\lim\limits_{\Delta x\to0}3x^2+3x\Delta x+(\Delta x)^2\\{}\\ =3x^2

## x^4的导数

\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}x^2=2x\\{}\\ \frac{\mathrm{d}}{\mathrm{d}x}x^3=3x^2

x^4的导数实际上是4x^3，你可以试着用差商公式推一遍。

\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}x^4=4x^3

## x^n的导数

\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}x^n=nx^{n-1}