Questions in category: 导数及微分 (Derivatives and differentials)

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## 1. 证明: $\cos x < \frac{1}{x}$, $\forall\ x\in(0,\frac{\pi}{2})$.

Posted by haifeng on 2022-11-10 13:22:39 last update 2022-11-10 13:22:54 | Answers (0) | 收藏

## 2. 证明: 当 $x > 0$ 时, $e^{\frac{x}{x+1}} < (1+\frac{1}{x})^x < e$.

Posted by haifeng on 2022-11-10 13:21:28 last update 2022-11-10 13:21:28 | Answers (0) | 收藏

$e^{\frac{x}{x+1}} < (1+\frac{1}{x})^x < e.$

## 3. 证明: 当 $x > 1$ 时, $\frac{\pi}{4} < x(\frac{\pi}{2}-\arctan x) < 1$.

Posted by haifeng on 2022-11-10 12:38:42 last update 2022-11-10 12:38:42 | Answers (1) | 收藏

$\frac{\pi}{4} < x(\frac{\pi}{2}-\arctan x) < 1.$

## 4. 设函数 $f(x)$ 定义在 $(0,+\infty)$ 上, 对任意 $x_1,x_2\in(0,+\infty)$, 有 $f(x_1\cdot x_2)=f(x_1)+f(x_2)$, 且 $f'(1)=1$, 证明: $f(x)=\ln x$.

Posted by haifeng on 2022-10-13 20:49:00 last update 2022-10-13 20:49:00 | Answers (1) | 收藏

## 5. 设 $f(x)$ 为可导函数, 求下列函数的导数.

Posted by haifeng on 2022-10-06 11:40:35 last update 2022-10-06 11:40:35 | Answers (1) | 收藏

$y=\sin(f(x))\cdot f(\sin x)$

## 6. 求函数 $y=\frac{\sqrt{x^2+1}+\sqrt{x^2-1}}{\sqrt{x^2+1}-\sqrt{x^2-1}}$ 的导数.

Posted by haifeng on 2022-10-06 11:39:06 last update 2022-10-06 11:39:06 | Answers (1) | 收藏

$y=\frac{\sqrt{x^2+1}+\sqrt{x^2-1}}{\sqrt{x^2+1}-\sqrt{x^2-1}}$

## 7. 设 $f(x)=a_1\sin x+a_2\sin 2x+\cdots+a_n\sin nx$, 且 $|f(x)|\leqslant |\sin x|$, 证明: $|a_1+2a_2+\cdots+na_n|\leqslant 1$.

Posted by haifeng on 2022-10-01 23:25:15 last update 2022-10-01 23:25:15 | Answers (0) | 收藏

$|a_1+2a_2+\cdots+na_n|\leqslant 1.$

## 8. 利用导数定义证明 $(\arcsin x)'=\frac{1}{\sqrt{1-x^2}}$.

Posted by haifeng on 2022-09-30 14:06:18 last update 2022-09-30 14:06:18 | Answers (1) | 收藏

## 9. 利用导数定义证明 $(\tan x)'=\sec^2 x$.

Posted by haifeng on 2022-09-30 13:57:51 last update 2022-09-30 14:01:10 | Answers (1) | 收藏

## 10. 如何设计多元函数偏导数的求解?

Posted by haifeng on 2022-09-24 16:07:30 last update 2022-09-24 16:07:30 | Answers (0) | 收藏

[Idea]

u=x^2+y^2,  确认 u=u(x,y) 是关于 x,y 的二元函数, 且 u(x,y)=x^2+y^2

v=xy,  确认 v=v(x,y) 是关于 x,y 的二元函数, 且 v(x,y)=xy

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