scalars.Rmd

# Arithmetic Functions

## Addition

$$
c = a + b
$$

### Derivatives {-}

$$
\frac{\partial}{\partial a} c = 1
\qquad
\frac{\partial}{\partial b} c = 1
$$

### Tangent {-}

$$
\dot{c} = \dot{a} + \dot{b}
$$

### Adjoints {-}

$$
 \overline{a} \ {+=} \ \overline{c}
\qquad
 \overline{b} \ {+=} \ \overline{c}
$$


## Subtraction

$$
c = a - b
$$

### Derivatives {-}

$$
\frac{\partial}{\partial a} c = 1
\qquad
\frac{\partial}{\partial b} c = -1
$$


### Tangent {-}

$$
\dot{c} = \dot{a} - \dot{b}
$$

### Adjoints {-}

$$
 \overline{a} {+=} \overline{c}
\qquad
 \overline{b} {+=} - \overline{c}
$$

## Multiplication

$$
c = a \cdot b
$$

### Derivatives {-}

$$
\frac{\partial}{\partial a} c = b
\qquad
\frac{\partial}{\partial b} c = a
$$


### Tangent {-}

$$
\dot{c} = \dot{a} \cdot b + \dot{b} \cdot a
$$

### Adjoints {-}

$$
 \overline{a} \ {+=} \ \overline{c} \cdot b
\qquad
 \overline{b} \ {+=} \ \overline{c} \cdot a
$$

## Division

$$
c = \frac{a}{b}
$$

### Derivatives {-}

$$
\frac{\partial}{\partial a} c = \frac{1}{b}
\qquad
\frac{\partial}{\partial b} c = - \frac{a}{b^2}
$$


### Tangent {-}

$$
\dot{c} = \frac{\dot{a}}{b} - \frac{\dot{b} \cdot a}{b^2}
$$

### Adjoints {-}

$$
\overline{a} \ {+=} \ \frac{\overline{c}}{b}
\qquad
\overline{b} \ {+=} \ - \frac{\overline{c} \cdot a}{b^2}
$$

## Exponential

$$
c = \exp(a)
$$

### Derivatives {-}

$$
\frac{\partial}{\partial a} c = \exp(a)
$$

### Tangent {-}

$$
\dot{c} = \dot{a} \cdot \exp(a)
$$

### Adjoints {-}

$$
\overline{a} \ {+=} \ \overline{c} \cdot \exp(a)
$$

## Exponential (base 2)

$$
c = 2^a
$$

### Derivatives {-}

$$
\frac{\partial}{\partial a} c =  \log(a) \cdot 2^a
$$

### Tangent {-}

$$
\dot{c} = \dot{a} \cdot \log(a) \cdot 2^a
$$

### Adjoints {-}

$$
\overline{a} \ {+=} \ \overline{c} \cdot \log(a) \cdot 2^a
$$


## Logarithm (base e)

$$
c = \log(a)
$$

### Derivatives {-}

$$
\frac{\partial}{\partial a} c = \frac{1}{a}
$$

### Tangent {-}

$$
\dot{c} = \frac{\dot{a}}{a}
$$

### Adjoints {-}

$$
\overline{a} \ {+=} \ \frac{\overline{c}}{a}
$$

## Logarithm (base 2)

$$
c = \log_2(a)
$$

### Derivatives {-}

$$
\frac{\partial}{\partial a} c = \frac{1}{a \cdot \log(2)}
$$

### Tangent {-}

$$
\dot{c} = \frac{\dot{a}}{a \cdot \log(2)}
$$

### Adjoints {-}

$$
\overline{a} \ {+=} \ \frac{\overline{c}}{a \cdot \log(2)}
$$

## Logarithm (base 10)

$$
c = \log_{10}(a)
$$

### Derivatives {-}

$$
\frac{\partial}{\partial a} c = \frac{1}{a \cdot \log(10)}
$$

### Tangent {-}

$$
\dot{c} = \frac{\dot{a}}{a \cdot \log(10)}
$$

### Adjoints {-}

$$
\overline{a} \ {+=} \ \frac{\overline{c}}{a \cdot \log(10)}
$$

## Power

$$
c = a^b
$$

### Derivatives {-}

$$
\frac{\partial}{\partial a} c = b \cdot a^{b-1}
\qquad
\frac{\partial}{\partial b} c = \log(a) \cdot a^b
$$

### Tangent {-}

$$
\dot{c} = \dot{a} \cdot b \cdot a^{b-1} + \dot{b} \cdot \log(a) \cdot a^b = \left( \dot{a} \cdot \frac{b}{a} + \dot{b} \cdot \log(a) \right) \cdot a^b
$$

### Adjoints {-}

$$
\overline{a} \ {+=} \ \overline{c} \cdot b \cdot a^{b-1} 
\qquad 
\overline{b} \ {+=} \ \overline{c} \cdot \log(a) \cdot a^b
$$

## Square

$$
c = a^2
$$

### Derivatives {-}

$$
\frac{\partial}{\partial a} c = 2a
$$

### Tangent {-}

$$
\dot{c} = \dot{a} \cdot 2a
$$

### Adjoints {-}

$$
\overline{a} \ {+=} \ \overline{c} \cdot 2a
$$

## Square root

$$
c = \sqrt{a}
$$

### Derivatives {-}

$$
\frac{\partial}{\partial a} c = -\frac{1}{2 \sqrt{a}}
$$

### Tangent {-}

$$
\dot{c} = -\frac{\dot{a}}{2 \sqrt{a}}
$$

### Adjoints {-}

$$
 \overline{a} \ {+=} \ -\frac{\overline{c}}{2 \sqrt{a}}
$$


## Inverse

$$
c = \frac{1}{a}
$$

### Derivatives {-}

$$
\frac{\partial}{\partial a} c = - \frac{1}{a^2}
$$

### Tangent {-}

$$
\dot{c} = - \frac{\dot{a}}{a^2}
$$

### Adjoints {-}

$$
 \overline{a} \ {+=} \ - \frac{\overline{c}}{a^2}
$$


## Cos

$$
c = \cos(a)
$$

### Derivatives {-}

$$
\frac{\partial}{\partial a} c = - \sin(a)
$$

### Tangent {-}

$$
\dot{c} = - \dot{a} \cdot \sin(a)
$$

### Adjoints {-}

$$
 \overline{a} \ {+=} \ - \overline{c} \cdot \sin(a)
$$

## Sin

$$
c = \sin(a)
$$

### Derivatives {-}

$$
\frac{\partial}{\partial a} c = \cos(a)
$$

### Tangent {-}

$$
\dot{c} = \dot{a} \cdot \cos(a)
$$

### Adjoints {-}

$$
\overline{a} \ {+=} \ \overline{c} \cdot \cos(a)
$$

## Tan

$$
c = \tan(a)
$$

### Derivatives {-}

$$
\frac{\partial}{\partial a} c = \frac{1}{\cos^2(a)}
$$

### Tangent {-}

$$
\dot{c} = \frac{\dot{a}}{\cos^2(a)}
$$

### Adjoints {-}

$$
\overline{a} \ {+=} \ \frac{\overline{c}}{\cos^2(a)}
$$

## Arccos

$$
c = \arccos(a)
$$

### Derivatives {-}

$$
\frac{\partial}{\partial a} c = - \frac{1}{\sqrt{1-a^2}}
$$

### Tangent {-}

$$
\dot{c} = - \frac{\dot{a}}{\sqrt{1-a^2}}
$$

### Adjoints {-}

$$
 \overline{a} \ {+=} \ - \frac{\overline{c}}{\sqrt{1-a^2}}
$$

## Arcsin

$$
c = \arcsin(a)
$$

### Derivatives {-}

$$
\frac{\partial}{\partial a} c = \frac{1}{\sqrt{1-a^2}}
$$

### Tangent {-}

$$
\dot{c} = \frac{\dot{a}}{\sqrt{1-a^2}}
$$

### Adjoints {-}

$$
 \overline{a} \ {+=} \ \frac{\overline{c}}{\sqrt{1-a^2}}
$$

## Arctan

$$
c = \arctan(a)
$$

### Derivatives {-}

$$
\frac{\partial}{\partial a} c = \frac{1}{1+a^2}
$$

### Tangent {-}

$$
\dot{c} = \frac{\dot{a}}{1+a^2}
$$

### Adjoints {-}

$$
 \overline{a} \ {+=} \ \frac{\overline{c}}{1+a^2}
$$


## Cosh

$$
c = \cosh(a)
$$

### Derivatives {-}

$$
\frac{\partial}{\partial a} c = \sinh(a)
$$

### Tangent {-}

$$
\dot{c} = \dot{a} \cdot \sinh(a)
$$

### Adjoints {-}

$$
 \overline{a} \ {+=} \ \overline{c} \cdot \sinh(a)
$$

## Sinh

$$
c = \sinh(a)
$$

### Derivatives {-}

$$
\frac{\partial}{\partial a} c = \cosh(a)
$$

### Tangent {-}

$$
\dot{c} = \dot{a} \cdot \cosh(a)
$$

### Adjoints {-}

$$
\overline{a} \ {+=} \ \overline{c} \cdot \cosh(a)
$$

## Tanh

$$
c = \tanh(a)
$$

### Derivatives {-}

$$
\frac{\partial}{\partial a} c = \frac{1}{\cosh^2(a)}
$$

### Tangent {-}

$$
\dot{c} = \frac{\dot{a}}{\cosh^2(a)}
$$

### Adjoints {-}

$$
\overline{a} \ {+=} \ \frac{\overline{c}}{\cosh^2(a)}
$$

## Arccosh

$$
c = \text{arccosh}(a)
$$

### Derivatives {-}

$$
\frac{\partial}{\partial a} c = \frac{1}{\sqrt{a^2-1}}
$$

### Tangent {-}

$$
\dot{c} = \frac{\dot{a}}{\sqrt{a^2-1}}
$$

### Adjoints {-}

$$
 \overline{a} \ {+=} \ \frac{\overline{c}}{\sqrt{a^2-1}}
$$

## Arcsinh

$$
c = \text{arcsinh}(a)
$$

### Derivatives {-}

$$
\frac{\partial}{\partial a} c = \frac{1}{\sqrt{1+a^2}}
$$

### Tangent {-}

$$
\dot{c} = \frac{\dot{a}}{\sqrt{1+a^2}}
$$

### Adjoints {-}

$$
 \overline{a} \ {+=} \ \frac{\overline{c}}{\sqrt{1+a^2}}
$$

## Arctanh

$$
c = \text{arctanh}(a)
$$

### Derivatives {-}

$$
\frac{\partial}{\partial a} c = \frac{1}{1-a^2}
$$

### Tangent {-}

$$
\dot{c} = \frac{\dot{a}}{1-a^2}
$$

### Adjoints {-}

$$
 \overline{a} \ {+=} \ \frac{\overline{c}}{1-a^2}
$$