万门大学高等数学特训班课前公式整理

川长思鸟来 2021-12-13 14:49 178阅读 0赞

# 一.等价无穷小关系式(x->0) #

![(1)sinx=x][1_sinx_x]

![(2) tanx=x][2_ tanx_x]

![(3)arcsinx=x][3_arcsinx_x]

![(4)1-cosx=\\frac\{1\}\{2\}x^\{2\}][4_1-cosx_frac_1_2_x_2]

![(5)\\sqrt\[n\]\{1+x\}-1=\\frac\{1\}\{n\}x][5_sqrt_n_1_x_-1_frac_1_n_x]

![(6)ln(1+x)=x][6_ln_1_x_x]

![(7)e^\{x\}-1=x][7_e_x_-1_x]

![(8)(1+x)^\{a\}-1=ax][8_1_x_a_-1_ax]

# 二.导数公式 #

![(1)(cos(x))^\{'\}=-sin(x)][1_cos_x_-sin_x]

![(2)(sin(x))^\{'\}=cos(x)][2_sin_x_cos_x]

![(3)(tan(x))^\{'\}=\\frac\{1\}\{cos^\{2\}(x)\}][3_tan_x_frac_1_cos_2_x]

![(4)(cot(x))^\{'\}=-\\frac\{1\}\{sin^\{2\}(x)\}][4_cot_x_-_frac_1_sin_2_x]

![(5)(a^\{x\})^\{'\}=a^\{x\}lna(a>0 ,a\\neq 1 )][5_a_x_a_x_lna_a_0 _a_neq 1]

![(6)(e^\{x\})^\{'\}=e^\{x\}][6_e_x_e_x]

![(7)(loga(x))^\{'\}=\\frac\{1\}\{xlna\}(a>0,a\\neq 1)][7_loga_x_frac_1_xlna_a_0_a_neq 1]

![(8)(lnx)^\{'\}=\\frac\{1\}\{x\}][8_lnx_frac_1_x]

![(9)(arcsin(x))^\{'\}=\\frac\{1\}\{\\sqrt\{1-x^\{2\}\}\}][9_arcsin_x_frac_1_sqrt_1-x_2]

![(10)(arccos(x))^\{'\}=-\\frac\{1\}\{\\sqrt\{1-x^\{2\}\}\}][10_arccos_x_-_frac_1_sqrt_1-x_2]

![(11)(arctan(x))^\{'\}=\\frac\{1\}\{1+x^\{2\}\}][11_arctan_x_frac_1_1_x_2]

![(12)(arccot(x))^\{'\}=-\\frac\{1\}\{1+x^\{2\}\}][12_arccot_x_-_frac_1_1_x_2]

# 三.微积分  #

![\\int\_\{a\}^\{b\}f^\{'\}(x)dx=f(x)\\left \\right |\_\{a\}^\{b\}=f(b)-f(a)][int_a_b_f_x_dx_f_x_left _right _a_b_f_b_-f_a]

# 参考资料  #

[高数总结][Link 1]

[1_sinx_x]: https://private.codecogs.com/gif.latex?%281%29sinx%3Dx
[2_ tanx_x]: https://private.codecogs.com/gif.latex?%282%29%20tanx%3Dx
[3_arcsinx_x]: https://private.codecogs.com/gif.latex?%283%29arcsinx%3Dx
[4_1-cosx_frac_1_2_x_2]: https://private.codecogs.com/gif.latex?%284%291-cosx%3D%5Cfrac%7B1%7D%7B2%7Dx%5E%7B2%7D
[5_sqrt_n_1_x_-1_frac_1_n_x]: https://private.codecogs.com/gif.latex?%285%29%5Csqrt%5Bn%5D%7B1&plus;x%7D-1%3D%5Cfrac%7B1%7D%7Bn%7Dx
[6_ln_1_x_x]: https://private.codecogs.com/gif.latex?%286%29ln%281&plus;x%29%3Dx
[7_e_x_-1_x]: https://private.codecogs.com/gif.latex?%287%29e%5E%7Bx%7D-1%3Dx
[8_1_x_a_-1_ax]: https://private.codecogs.com/gif.latex?%288%29%281&plus;x%29%5E%7Ba%7D-1%3Dax
[1_cos_x_-sin_x]: https://private.codecogs.com/gif.latex?%281%29%28cos%28x%29%29%5E%7B%27%7D%3D-sin%28x%29
[2_sin_x_cos_x]: https://private.codecogs.com/gif.latex?%282%29%28sin%28x%29%29%5E%7B%27%7D%3Dcos%28x%29
[3_tan_x_frac_1_cos_2_x]: https://private.codecogs.com/gif.latex?%283%29%28tan%28x%29%29%5E%7B%27%7D%3D%5Cfrac%7B1%7D%7Bcos%5E%7B2%7D%28x%29%7D
[4_cot_x_-_frac_1_sin_2_x]: https://private.codecogs.com/gif.latex?%284%29%28cot%28x%29%29%5E%7B%27%7D%3D-%5Cfrac%7B1%7D%7Bsin%5E%7B2%7D%28x%29%7D
[5_a_x_a_x_lna_a_0 _a_neq 1]: https://private.codecogs.com/gif.latex?%285%29%28a%5E%7Bx%7D%29%5E%7B%27%7D%3Da%5E%7Bx%7Dlna%28a%3E0%20%2Ca%5Cneq%201%20%29
[6_e_x_e_x]: https://private.codecogs.com/gif.latex?%286%29%28e%5E%7Bx%7D%29%5E%7B%27%7D%3De%5E%7Bx%7D
[7_loga_x_frac_1_xlna_a_0_a_neq 1]: https://private.codecogs.com/gif.latex?%287%29%28loga%28x%29%29%5E%7B%27%7D%3D%5Cfrac%7B1%7D%7Bxlna%7D%28a%3E0%2Ca%5Cneq%201%29
[8_lnx_frac_1_x]: https://private.codecogs.com/gif.latex?%288%29%28lnx%29%5E%7B%27%7D%3D%5Cfrac%7B1%7D%7Bx%7D
[9_arcsin_x_frac_1_sqrt_1-x_2]: https://private.codecogs.com/gif.latex?%289%29%28arcsin%28x%29%29%5E%7B%27%7D%3D%5Cfrac%7B1%7D%7B%5Csqrt%7B1-x%5E%7B2%7D%7D%7D
[10_arccos_x_-_frac_1_sqrt_1-x_2]: https://private.codecogs.com/gif.latex?%2810%29%28arccos%28x%29%29%5E%7B%27%7D%3D-%5Cfrac%7B1%7D%7B%5Csqrt%7B1-x%5E%7B2%7D%7D%7D
[11_arctan_x_frac_1_1_x_2]: https://private.codecogs.com/gif.latex?%2811%29%28arctan%28x%29%29%5E%7B%27%7D%3D%5Cfrac%7B1%7D%7B1&plus;x%5E%7B2%7D%7D
[12_arccot_x_-_frac_1_1_x_2]: https://private.codecogs.com/gif.latex?%2812%29%28arccot%28x%29%29%5E%7B%27%7D%3D-%5Cfrac%7B1%7D%7B1&plus;x%5E%7B2%7D%7D
[int_a_b_f_x_dx_f_x_left _right _a_b_f_b_-f_a]: https://private.codecogs.com/gif.latex?%5Cint_%7Ba%7D%5E%7Bb%7Df%5E%7B%27%7D%28x%29dx%3Df%28x%29%5Cleft%20%5Cright%20%7C_%7Ba%7D%5E%7Bb%7D%3Df%28b%29-f%28a%29
[Link 1]: https://jingyan.baidu.com/article/fec7a1e5d182091191b4e753.html