文章目录
- 1、激活函数的实现
- 1.1 sigmoid
- 1.1.1 函数
- 1.1.2 导数
- 1.1.3 代码实现
- 1.2 softmax
- 1.2.1 函数
- 1.2.2 导数
- 1.2.3 代码实现
- 1.3 tanh
- 1.3.1 函数
- 1.3.2 导数
- 1.3.3 代码实现
- 1.4 relu
- 1.4.1 函数
- 1.4.2 导数
- 1.4.3 代码实现
- 1.5 leakyrelu
- 1.5.1 函数
- 1.5.2 导数
- 1.5.3 代码实现
- 1.6 ELU
- 1.61 函数
- 1.6.2 导数
- 1.6.3 代码实现
- 1.7 selu
- 1.7.1 函数
- 1.7.2 导数
- 1.7.3 代码实现
- 1.8 softplus
- 1.81 函数
- 1.8.2 导数
- 1.8.3 代码实现
- 1.9 Swish
- 1.9.1 函数
- 1.9.2 导数
- 1.9.3 代码实现
- 1.10 Mish
- 1.10.1 函数
- 1.10.2 导数
- 1.10.3 代码实现
- 1.11 SiLU
- 1.11.1 函数
- 1.11.2 导数
- 1.11.3 代码实现
- 1.12 完整代码
1、激活函数的实现
1.1 sigmoid
1.1.1 函数
函数： f ( x ) = 1 1 + e − x f(x)=\frac{1}{1+e^{-x}} f(x)=1+e−x1
1.1.2 导数
求导过程：
根据： ( u v ) ′ = u ′ v − u v ′ v 2 \left ( \frac{u}{v} \right ){}'=\frac{{u}'v-u{v}'}{v^{2}} (vu)′=v2u′v−uv′
f ( x ) ′ = ( 1 1 + e − x ) ′ = 1 ′ × ( 1 + e − x ) − 1 × ( 1 + e − x ) ′ ( 1 + e − x ) 2 = e − x ( 1 + e − x ) 2 = 1 + e − x − 1 ( 1 + e − x ) 2 = ( 1 1 + e − x ) ( 1 − 1 1 + e − x ) = f ( x ) ( 1 − f ( x ) ) \begin{aligned} f(x)^{\prime} &=\left(\frac{1}{1+e^{-x}}\right)^{\prime} \\ &=\frac{1^{\prime} \times\left(1+e^{-x}\right)-1 \times\left(1+e^{-x}\right)^{\prime}}{\left(1+e^{-x}\right)^{2}} \\ &=\frac{e^{-x}}{\left(1+e^{-x}\right)^{2}} \\ &=\frac{1+e^{-x}-1}{\left(1+e^{-x}\right)^{2}} \\ &=\left(\frac{1}{1+e^{-x}}\right)\left(1-\frac{1}{1+e^{-x}}\right) \\ &=\quad f(x)(1-f(x)) \end{aligned} f(x)′=(1+e−x1)′=(1+e−x)21′×(1+e−x)−1×(1+e−x)′=(1+e−x)2e−x=(1+e−x)21+e−x−1=(1+e−x1)(1−1+e−x1)=f(x)(1−f(x))
1.1.3 代码实现
import numpy as np
class Sigmoid():
def __call__(self, x):
return 1 / (1 + np.exp(-x))
def gradient(self, x):
return self.__call__(x) * (1 - self.__call__(x))1.2 softmax
1.2.1 函数
softmax用于多分类过程中，它将多个神经元的输出，映射到（0,1）区间内，可以看成概率来理解，从而来进行多分类！
假设我们有一个数组，V，Vi表示V中的第i个元素，那么这个元素的softmax值就是:
S i = e i ∑ j e j S_{i}=\frac{e^{i}}{\sum _{j}e^{j}} Si=∑jejei
更形象的如下图表示：
y 1 = e z 1 e z 1 + e z 2 + e z 3 y 2 = e z 2 e z 1 + e z 2 + e z 3 y 3 = e z 3 e z 1 + e z 2 + e z 3 (1) y1=\frac{e^{z_{1}}}{e^{z_{1}}+e^{z_{2}}+e^{z_{3}}}\\ y2=\frac{e^{z_{2}}}{e^{z_{1}}+e^{z_{2}}+e^{z_{3}}}\\ y3=\frac{e^{z_{3}}}{e^{z_{1}}+e^{z_{2}}+e^{z_{3}}}\\ \tag{1} y1=ez1+ez2+ez3ez1y2=ez1+ez2+ez3ez2y3=ez1+ez2+ez3ez3(1)
要使用梯度下降，就需要一个损失函数，一般使用交叉熵作为损失函数，交叉熵函数形式如下：
L o s s = − ∑ i y i l n a i (2) Loss = -\sum_{i}^{}{y_{i}lna_{i} } \tag{2} Loss=−i∑yilnai(2)
1.2.2 导数
求导分为两种情况。
第一种j=i：
S i = e i ∑ j e j = e i ∑ i e i S_{i}=\frac{e^{i}}{\sum _{j}e^{j}}=\frac{e^{i}}{\sum _{i}e^{i}} Si=∑jejei=∑ieiei
推导过程如下：
f ′ = ( e i ∑ i e i ) ′ = ( e i ) × ∑ i e i − e i × e i ( ∑ i e i ) 2 = e i ∑ i e i − e i ∑ i e i × e i ∑ i e i = e i ∑ i e i ( 1 − e i ∑ i e i ) = f ( 1 − f ) \begin{aligned} f^{\prime}&=\left(\frac{e^{i}}{\sum_{i} e^{i}}\right)^{\prime} & \\ &=\frac{\left(e^{i}\right) \times \sum_{i} e^{i}-e^{i} \times e^{i}}{\left(\sum_{i} e^{i}\right)^{2}} \\ &=\frac{e^{i}}{\sum_{i} e^{i}}-\frac{e^{i}}{\sum_{i} e^{i}} \times \frac{e^{i}}{\sum_{i} e^{i}} \\ &= \frac{e^{i}}{\sum_{i} e^{i}}\left(1-\frac{e^{i}}{\sum_{i} e^{i}}\right) \\ &= f(1-f) \end{aligned} f′=(∑ieiei)′=(∑iei)2(ei)×∑iei−ei×ei=∑ieiei−∑ieiei×∑ieiei=∑ieiei(1−∑ieiei)=f(1−f)
1.2.3 代码实现
import numpy as np
class Softmax():
def __call__(self, x):
e_x = np.exp(x - np.max(x, axis=-1, keepdims=True))
return e_x / np.sum(e_x, axis=-1, keepdims=True)
def gradient(self, x):
p = self.__call__(x)
return p * (1 - p)1.3 tanh
1.3.1 函数
t a n h ( x ) = e x − e − x e x + e − x tanh(x)=\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}} tanh(x)=ex+e−xex−e−x
1.3.2 导数
求导过程：
tanh ( x ) ′ = ( e x − e − x e x + e − x ) ′ = ( e x − e − x ) ′ ( e x + e − x ) − ( e x − e − x ) ( e x + e − x ) ′ ( e x + e − x ) 2 = ( e x + e − x ) 2 − ( e x ⋅ e − x ) 2 ( e x + e − x ) 2 = 1 − ( e x − e − x e x + e − x ) 2 = 1 − tanh ( x ) 2 \begin{aligned} \tanh (x)^{\prime} &=\left(\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}\right)^{\prime} \\ &=\frac{\left(e^{x}-e^{-x}\right)^{\prime}\left(e^{x}+e^{-x}\right)-\left(e^{x}-e^{-x}\right)\left(e^{x}+e^{-x}\right)^{\prime}}{\left(e^{x}+e^{-x}\right)^{2}} \\ &=\frac{\left(e^{x}+e^{-x}\right)^{2}-\left(e^{x} \cdot e^{-x}\right)^{2}}{\left(e^{x}+e^{-x}\right)^{2}} \\ &=1-\left(\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}\right)^{2} \\ &=1-\tanh (x)^{2} \end{aligned} tanh(x)′=(ex+e−xex−e−x)′=(ex+e−x)2(ex−e−x)′(ex+e−x)−(ex−e−x)(ex+e−x)′=(ex+e−x)2(ex+e−x)2−(ex⋅e−x)2=1−(ex+e−xex−e−x)2=1−tanh(x)2
1.3.3 代码实现
import numpy as np
class TanH():
def __call__(self, x):
return 2 / (1 + np.exp(-2*x)) - 1
def gradient(self, x):
return 1 - np.power(self.__call__(x), 2)1.4 relu
1.4.1 函数
f ( x ) = max ( 0 , x ) f(x)=\max (0, x) f(x)=max(0,x)
1.4.2 导数
f ′ ( x ) = { 1 if ( x > 0 ) 0 if ( x < = 0 ) f^{\prime}(x)=\left\{\begin{array}{cc} 1 & \text { if } (x>0) \\ 0 & \text { if } (x<=0) \end{array}\right. f′(x)={10 if (x>0) if (x<=0)
1.4.3 代码实现
import numpy as np
class ReLU():
def __call__(self, x):
return np.where(x >= 0, x, 0)
def gradient(self, x):
return np.where(x >= 0, 1, 0)1.5 leakyrelu
1.5.1 函数
f ( x ) = max ( a x , x ) f(x)=\max (a x, x) f(x)=max(ax,x)
1.5.2 导数
f ′ ( x ) = { 1 if ( x > 0 ) a if ( x < = 0 ) f^{\prime}(x)=\left\{\begin{array}{cl} 1 & \text { if } (x>0) \\ a & \text { if }(x<=0) \end{array}\right. f′(x)={1a if (x>0) if (x<=0)
1.5.3 代码实现
import numpy as np
class LeakyReLU():
def __init__(self, alpha=0.2):
self.alpha = alpha
def __call__(self, x):
return np.where(x >= 0, x, self.alpha * x)
def gradient(self, x):
return np.where(x >= 0, 1, self.alpha)1.6 ELU
1.61 函数
f ( x ) = { x , if x ≥ 0 a ( e x − 1 ) , if ( x < 0 ) f(x)=\left\{\begin{array}{cll} x, & \text { if } x \geq 0 \\ a\left(e^{x}-1\right), & \text { if } (x<0) \end{array}\right. f(x)={x,a(ex−1), if x≥0 if (x<0)
1.6.2 导数
当x>=0时，导数为1。
当x<0时，导数的推导过程：
f ( x ) ′ = ( a ( e x − 1 ) ) ′ = a e x = a ( e x − 1 ) + a = f ( x ) + a = a e x \begin{aligned} \\ f(x)^{\prime} &=\left(a\left(e^{x}-1\right)\right)^{\prime} \\ &=a e^{x} \\ &\left.=a (e^{x}-1\right)+a \\ &=f(x)+a=ae^{x} \end{aligned} f(x)′=(a(ex−1))′=aex=a(ex−1)+a=f(x)+a=aex
所以，完整的导数为：
f ′ = { 1 if x ≥ 0 f ( x ) + a = a e x if x < 0 f^{\prime}=\left\{\begin{array}{cll} 1 & \text { if } & x \geq 0 \\ f(x)+a=ae^{x} & \text { if } & x<0 \end{array}\right. f′={1f(x)+a=aex if if x≥0x<0
1.6.3 代码实现
import numpy as np
class ELU():
def __init__(self, alpha=0.1):
self.alpha = alpha
def __call__(self, x):
return np.where(x >= 0.0, x, self.alpha * (np.exp(x) - 1))
def gradient(self, x):
return np.where(x >= 0.0, 1, self.__call__(x) + self.alpha)1.7 selu
1.7.1 函数
selu ( x ) = λ { x if ( x > 0 ) α e x − α if ( x ⩽ 0 ) \operatorname{selu}(x)=\lambda \begin{cases}x & \text { if } (x>0) \\ \alpha e^{x}-\alpha & \text { if } (x \leqslant 0)\end{cases} selu(x)=λ{xαex−α if (x>0) if (x⩽0)
1.7.2 导数
selu ′ ( x ) = λ { 1 x > 0 α e x ⩽ 0 \operatorname{selu}^{\prime}(x)=\lambda \begin{cases}1 & x>0 \\ \alpha e^{x} & \leqslant 0\end{cases} selu′(x)=λ{1αexx>0⩽0
1.7.3 代码实现
import numpy as np
class SELU():
# Reference : https://arxiv.org/abs/1706.02515,
# https://github.com/bioinf-jku/SNNs/blob/master/SelfNormalizingNetworks_MLP_MNIST.ipynb
def __init__(self):
self.alpha = 1.6732632423543772848170429916717
self.scale = 1.0507009873554804934193349852946
def __call__(self, x):
return self.scale * np.where(x >= 0.0, x, self.alpha*(np.exp(x)-1))
def gradient(self, x):
return self.scale * np.where(x >= 0.0, 1, self.alpha * np.exp(x))1.8 softplus
1.81 函数
Softplus ( x ) = log ( 1 + e x ) \operatorname{Softplus}(x)=\log \left(1+e^{x}\right) Softplus(x)=log(1+ex)
1.8.2 导数
log默认的底数是 e e e
f ′ ( x ) = e x ( 1 + e x ) ln e = 1 1 + e − x = σ ( x ) f^{\prime}(x)=\frac{e^{x}}{(1+e^{x})\ln e}=\frac{1}{1+e^{-x}}=\sigma(x) f′(x)=(1+ex)lneex=1+e−x1=σ(x)
1.8.3 代码实现
import numpy as np
class SoftPlus():
def __call__(self, x):
return np.log(1 + np.exp(x))
def gradient(self, x):
return 1 / (1 + np.exp(-x))1.9 Swish
1.9.1 函数
f ( x ) = x ⋅ sigmoid ( β x ) f(x)=x \cdot \operatorname{sigmoid}(\beta x) f(x)=x⋅sigmoid(βx)
1.9.2 导数
f ′ ( x ) = σ ( β x ) + β x ⋅ σ ( β x ) ( 1 − σ ( β x ) ) = σ ( β x ) + β x ⋅ σ ( β x ) − β x ⋅ σ ( β x ) 2 = β x ⋅ σ ( x ) + σ ( β x ) ( 1 − β x ⋅ σ ( β x ) ) = β f ( x ) + σ ( β x ) ( 1 − β f ( x ) ) \begin{aligned} f^{\prime}(x) &=\sigma(\beta x)+\beta x \cdot \sigma(\beta x)(1-\sigma(\beta x)) \\ &=\sigma(\beta x)+\beta x \cdot \sigma(\beta x)-\beta x \cdot \sigma(\beta x)^{2} \\ &=\beta x \cdot \sigma(x)+\sigma(\beta x)(1-\beta x \cdot \sigma(\beta x)) \\ &=\beta f(x)+\sigma(\beta x)(1-\beta f(x)) \end{aligned} f′(x)=σ(βx)+βx⋅σ(βx)(1−σ(βx))=σ(βx)+βx⋅σ(βx)−βx⋅σ(βx)2=βx⋅σ(x)+σ(βx)(1−βx⋅σ(βx))=βf(x)+σ(βx)(1−βf(x))
1.9.3 代码实现
import numpy as np
class Swish(object):
def __init__(self, b):
self.b = b
def __call__(self, x):
return x * (np.exp(self.b * x) / (np.exp(self.b * x) + 1))
def gradient(self, x):
return self.b * x / (1 + np.exp(-self.b * x)) + (1 / (1 + np.exp(-self.b * x)))(
1 - self.b * (x / (1 + np.exp(-self.b * x))))1.10 Mish
1.10.1 函数
f ( x ) = x ∗ tanh ( ln ( 1 + e x ) ) f(x)=x * \tanh \left(\ln \left(1+e^{x}\right)\right) f(x)=x∗tanh(ln(1+ex))
1.10.2 导数
f ′ ( x ) = sech 2 ( soft plus ( x ) ) x sigmoid ( x ) + f ( x ) x = Δ ( x ) s w i sh ( x ) + f ( x ) x \begin{gathered} f^{\prime}(x)=\operatorname{sech}^{2}(\operatorname{soft} \operatorname{plus}(x)) x \operatorname{sigmoid}(x)+\frac{f(x)}{x} \\ =\Delta(x) s w i \operatorname{sh}(x)+\frac{f(x)}{x} \end{gathered} f′(x)=sech2(softplus(x))xsigmoid(x)+xf(x)=Δ(x)swish(x)+xf(x)
where softplus ( x ) = ln ( 1 + e x ) (x)=\ln \left(1+e^{x}\right) (x)=ln(1+ex) and sigmoid ( x ) = 1 / ( 1 + e − x ) (x)=1 /\left(1+e^{-x}\right) (x)=1/(1+e−x).
1.10.3 代码实现
import numpy as np
def sech(x):
"""sech函数"""
return 2 / (np.exp(x) + np.exp(-x))
def sigmoid(x):
"""sigmoid函数"""
return 1 / (1 + np.exp(-x))
def soft_plus(x):
"""softplus函数"""
return np.log(1 + np.exp(x))
def tan_h(x):
"""tanh函数"""
return (np.exp(x) - np.exp(-x)) / (np.exp(x) + np.exp(-x))
class Mish:
def __call__(self, x):
return x * tan_h(soft_plus(x))
def gradient(self, x):
return sech(soft_plus(x)) * sech(soft_plus(x)) * x * sigmoid(x) + tan_h(soft_plus(x))1.11 SiLU
1.11.1 函数
f ( x ) = x × s i g m o i d ( x ) f(x)=x \times sigmoid (x) f(x)=x×sigmoid(x)
1.11.2 导数
推导过程
f ( x ) ′ = ( x ⋅ s i g m o i d ( x ) ) ′ = s i g m o i d ( x ) + x ⋅ ( s i g m o i d ( x ) ( 1 − s i g m o i d ( x ) ) = s i g m o i d ( x ) + x ⋅ s i g m o i d ( x ) − x ⋅ s i g m o i d 2 ( x ) = f ( x ) + sigmoid ( x ) ( 1 − f ( x ) ) \begin{aligned} &f(x)^{\prime}=(x \cdot sigmoid(x))^{\prime}\\ &=sigmoid(x)+x \cdot(sigmoid(x)(1-sigmoid(x))\\ &=sigmoid(x)+x \cdot sigmoid(x)-x \cdot sigmoid^{2}(x)\\ &=f(x)+\operatorname{sigmoid}(x)(1-f(x)) \end{aligned} f(x)′=(x⋅sigmoid(x))′=sigmoid(x)+x⋅(sigmoid(x)(1−sigmoid(x))=sigmoid(x)+x⋅sigmoid(x)−x⋅sigmoid2(x)=f(x)+sigmoid(x)(1−f(x))
1.11.3 代码实现
import numpy as np
def sigmoid(x):
"""sigmoid函数"""
return 1 / (1 + np.exp(-x))
class SILU(object):
def __call__(self, x):
return x * sigmoid(x)
def gradient(self, x):
return self.__call__(x) + sigmoid(x) * (1 - self.__call__(x))1.12 完整代码
定义一个activation_function.py,将下面的代码复制进去，到这里激活函数就完成了。
import numpy as np
# Collection of activation functions
# Reference: https://en.wikipedia.org/wiki/Activation_function
class Sigmoid():
def __call__(self, x):
return 1 / (1 + np.exp(-x))
def gradient(self, x):
return self.__call__(x) * (1 - self.__call__(x))
class Softmax():
def __call__(self, x):
e_x = np.exp(x - np.max(x, axis=-1, keepdims=True))
return e_x / np.sum(e_x, axis=-1, keepdims=True)
def gradient(self, x):
p = self.__call__(x)
return p * (1 - p)
class TanH():
def __call__(self, x):
return 2 / (1 + np.exp(-2 * x)) - 1
def gradient(self, x):
return 1 - np.power(self.__call__(x), 2)
class ReLU():
def __call__(self, x):
return np.where(x >= 0, x, 0)
def gradient(self, x):
return np.where(x >= 0, 1, 0)
class LeakyReLU():
def __init__(self, alpha=0.2):
self.alpha = alpha
def __call__(self, x):
return np.where(x >= 0, x, self.alpha * x)
def gradient(self, x):
return np.where(x >= 0, 1, self.alpha)
class ELU(object):
def __init__(self, alpha=0.1):
self.alpha = alpha
def __call__(self, x):
return np.where(x >= 0.0, x, self.alpha * (np.exp(x) - 1))
def gradient(self, x):
return np.where(x >= 0.0, 1, self.__call__(x) + self.alpha)
class SELU():
# Reference : https://arxiv.org/abs/1706.02515,
# https://github.com/bioinf-jku/SNNs/blob/master/SelfNormalizingNetworks_MLP_MNIST.ipynb
def __init__(self):
self.alpha = 1.6732632423543772848170429916717
self.scale = 1.0507009873554804934193349852946
def __call__(self, x):
return self.scale * np.where(x >= 0.0, x, self.alpha * (np.exp(x) - 1))
def gradient(self, x):
return self.scale * np.where(x >= 0.0, 1, self.alpha * np.exp(x))
class SoftPlus(object):
def __call__(self, x):
return np.log(1 + np.exp(x))
def gradient(self, x):
return 1 / (1 + np.exp(-x))
class Swish(object):
def __init__(self, b):
self.b = b
def __call__(self, x):
return x * (np.exp(self.b * x) / (np.exp(self.b * x) + 1))
def gradient(self, x):
return self.b * x / (1 + np.exp(-self.b * x)) + (1 / (1 + np.exp(-self.b * x)))(
1 - self.b * (x / (1 + np.exp(-self.b * x))))
def sech(x):
"""sech函数"""
return 2 / (np.exp(x) + np.exp(-x))
def sigmoid(x):
"""sigmoid函数"""
return 1 / (1 + np.exp(-x))
def soft_plus(x):
"""softplus函数"""
return np.log(1 + np.exp(x))
def tan_h(x):
"""tanh函数"""
return (np.exp(x) - np.exp(-x)) / (np.exp(x) + np.exp(-x))
class Mish:
def __call__(self, x):
return x * tan_h(soft_plus(x))
def gradient(self, x):
return sech(soft_plus(x)) * sech(soft_plus(x)) * x * sigmoid(x) + tan_h(soft_plus(x))
class SILU(object):
def __call__(self, x):
return x * sigmoid(x)
def gradient(self, x):
return self.__call__(x) + sigmoid(x) * (1 - self.__call__(x))参考公式
( C ) ′ = 0 (C)^{\prime}=0 (C)′=0
( a x ) ′ = a x ln a \left(a^{x}\right)^{\prime}=a^{x} \ln a (ax)′=axlna
( x μ ) ′ = μ x μ − 1 \left(x^{\mu}\right)^{\prime}=\mu x^{\mu-1} (xμ)′=μxμ−1
( e x ) ′ = e x \left(e^{x}\right)^{\prime}=e^{x} (ex)′=ex
( sin x ) ′ = cos x (\sin x)^{\prime}=\cos x (sinx)′=cosx
( log a x ) ′ = 1 x ln a \left(\log _{a} x\right)^{\prime}=\frac{1}{x \ln a} (logax)′=xlna1
( cos x ) ′ = − sin x (\cos x)^{\prime}=-\sin x (cosx)′=−sinx
( ln x ) ′ = 1 x (\ln x)^{\prime}=\frac{1}{x} (lnx)′=x1
( tan x ) ′ = sec 2 x (\tan x)^{\prime}=\sec ^{2} x (tanx)′=sec2x
( arcsin x ) ′ = 1 1 − x 2 (\arcsin x)^{\prime}=\frac{1}{\sqrt{1-x^{2}}} (arcsinx)′=1−x2 1
( cot x ) ′ = − csc 2 x (\cot x)^{\prime}=-\csc ^{2} x (cotx)′=−csc2x
( arccos x ) ′ = − 1 1 − x 2 (\arccos x)^{\prime}=-\frac{1}{\sqrt{1-x^{2}}} (arccosx)′=−1−x2 1
( sec x ) ′ = sec x ⋅ tan x (\sec x)^{\prime}=\sec x \cdot \tan x (secx)′=secx⋅tanx
( arctan x ) ′ = 1 1 + x 2 (\arctan x)^{\prime}=\frac{1}{1+x^{2}} (arctanx)′=1+x21
( csc x ) ′ = − csc x ⋅ cot x (\csc x)^{\prime}=-\csc x \cdot \cot x (cscx)′=−cscx⋅cotx
( arccot x ) ′ = − 1 1 + x 2 (\operatorname{arccot} x)^{\prime}=-\frac{1}{1+x^{2}} (arccotx)′=−1+x21
双曲正弦: sinh x = e x − e − x 2 \sinh x=\frac{e^{x}-e^{-x}}{2} sinhx=2ex−e−x
双曲余弦: cosh x = e x + e − x 2 \cosh x=\frac{e^{x}+e^{-x}}{2} coshx=2ex+e−x
双曲正切: tanh x = sinh x cosh x = e x − e − x e x + e − x \tanh x=\frac{\sinh x}{\cosh x}=\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}} tanhx=coshxsinhx=ex+e−xex−e−x
双曲余切: coth x = 1 tanh x = e x + e − x e x − e − x \operatorname{coth} x=\frac{1}{\tanh x}=\frac{e^{x}+e^{-x}}{e^{x}-e^{-x}} cothx=tanhx1=ex−e−xex+e−x
双曲正割： sech x = 1 cosh x = 2 e x + e − x \operatorname{sech} x=\frac{1}{\cosh x}=\frac{2}{e^{x}+e^{-x}} sechx=coshx1=ex+e−x2
双曲余割： csch x = 1 sinh x = 2 e x − e − x \operatorname{csch} x=\frac{1}{\sinh x}=\frac{2}{e^{x}-e^{-x}} cschx=sinhx1=ex−e−x2
