用数学归纳法证明，对于所有正整数 $n$，

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。

用乘积法则验证第一阶导数，再在归纳步骤把指数函数提出，合并多项式系数。

For $n=1$,

which agrees with the given formula, so the result is true.

Assume that for some positive integer $k$,

Differentiating using the product rule,

This is the required expression for $n=k+1$. Since it holds for $n=1$ and truth for $n=k$ implies truth for $n=k+1$, the statement follows by mathematical induction for all positive integers $n$.