用数学归纳法证明：对每一个整数 $n\geqslant2$，

Answer all questions. You should use a calculator where appropriate.

You must show all necessary working clearly; no marks will be given for unsupported answers from a calculator.

Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless a different level of accuracy is specified in the question.

请回答所有题目。适当时应使用计算器。必须清楚展示所有必要的计算过程；仅给出计算器答案而没有过程，不予评分。除非题目另有精度要求，非精确数值答案保留 3 位有效数字，以度为单位的角度保留 1 位小数。

从n=2验证，不从n=1开始。归纳步骤再求一次导数，指数带出的负因子同时改变符号并补全阶乘。

Let $f(x)=x\ln x$, where $x>0$.

For $n=2$,

This agrees with $(-1)^2(2-2)!x^{1-2}=x^{-1}$, so the result is true for $n=2$.

Assume that, for some integer $k\geqslant2$,

Differentiating both sides,

Thus truth for $n=k$ implies truth for $n=k+1$. Since the result holds for $n=2$, by mathematical induction it holds for every integer $n\geqslant2$.