The function `derivative`

performs high-order symbolic and numerical differentiation for generic tensors with respect to an arbitrary number of variables. The function behaves differently depending on the arguments `order`

, the order of differentiation, and `var`

, the variable names with respect to which the derivatives are computed.

When multiple variables are provided and `order`

is a single integer \(n\), then the \(n\)-th order derivative is computed for each element of the tensor with respect to each variable:

\[D = \partial^{(n)} \otimes F\]

that is:

\[D_{i,\dots,j,k} = \partial^{(n)}_{k} F_{i,\dots,j}\]

where \(F\) is the tensor of functions and \(\partial_k^{(n)}\) denotes the \(n\)-th order partial derivative with respect to the \(k\)-th variable.

When `order`

matches the length of `var`

, it is assumed that the differentiation order is provided for each variable. In this case, each element is derived \(n_k\) times with respect to the \(k\)-th variable, for each of the \(m\) variables.

\[D_{i,\dots,j} = \partial^{(n_1)}_1\cdots\partial^{(n_m)}_m F_{i,\dots,j}\]

The same applies when `order`

is a named vector giving the differentiation order for each variable. For example, `order = c(x=1, y=2)`

differentiates once with respect to \(x\) and twice with respect to \(y\). A call with `order = c(x=1, y=0)`

is equivalent to `order = c(x=1)`

.

To compute numerical derivatives or to evaluate symbolic derivatives at a point, the function accepts a named vector for the argument `var`

; e.g. `var = c(x=1, y=2)`

evaluates the derivatives in \(x=1\) and \(y=2\). For `functions`

where the first argument is used as a parameter vector, `var`

should be a `numeric`

vector indicating the point at which the derivatives are to be calculated.

Symbolic derivatives of univariate functions: \(\partial_x sin(x)\).

```
derivative(f = "sin(x)", var = "x")
#> [1] "cos(x)"
```

Evaluation of symbolic and numerical derivatives: \(\partial_x sin(x)|_{x=0}\).

```
sym <- derivative(f = "sin(x)", var = c(x = 0))
num <- derivative(f = function(x) sin(x), var = c(x = 0))
```

```
#> Symbolic Numeric
#> 1 1
```

High order symbolic and numerical derivatives: \(\partial^{(4)}_x sin(x)|_{x=0}\).

```
sym <- derivative(f = "sin(x)", var = c(x = 0), order = 4)
num <- derivative(f = function(x) sin(x), var = c(x = 0), order = 4)
```

```
#> Symbolic Numeric
#> 0.000000e+00 -2.922023e-11
```

Symbolic derivatives of multivariate functions: \(\partial_x^{(1)}\partial_y^{(2)} y^2sin(x)\).

```
derivative(f = "y^2*sin(x)", var = c("x", "y"), order = c(1, 2))
#> [1] "2 * cos(x)"
```

Numerical derivatives of multivariate functions: \(\partial_x^{(1)}\partial_y^{(2)} y^2sin(x)|_{x=0,y=0}\) with degree of accuracy \(O(h^6)\).

```
f <- function(x, y) y^2*sin(x)
derivative(f, var = c(x=0, y=0), order = c(1, 2), accuracy = 6)
#> [1] 2
```

Symbolic gradient of multivariate functions: \(\partial_{x,y}x^2y^2\).

```
derivative("x^2*y^2", var = c("x", "y"))
#> [,1] [,2]
#> [1,] "2 * x * y^2" "x^2 * (2 * y)"
```

High order derivatives of multivariate functions: \(\partial^{(6)}_{x,y}x^6y^6\).

```
derivative("x^6*y^6", var = c("x", "y"), order = 6)
#> [,1] [,2]
#> [1,] "6 * (5 * (4 * (3 * 2))) * y^6" "x^6 * (6 * (5 * (4 * (3 * 2))))"
```

Numerical gradient of multivariate functions: \(\partial_{x,y}x^2y^2|_{x = 1, y = 2}\).

```
f <- function(x, y) x^2*y^2
derivative(f, var = c(x=1, y=2))
#> [,1] [,2]
#> [1,] 8 4
```

Numerical Jacobian of vector valued functions: \(\partial_{x,y}[xy,x^2y^2]|_{x = 1, y = 2}\).

```
f <- function(x, y) c(x*y, x^2*y^2)
derivative(f, var = c(x=1, y=2))
#> [,1] [,2]
#> [1,] 2 1
#> [2,] 8 4
```

Numerical Jacobian of vector valued where the first argument is used as a parameter vector: \(\partial_{X}[\sum_ix_i, \prod_ix_i]|_{X = 0}\).

```
f <- function(x) c(sum(x), prod(x))
derivative(f, var = c(0, 0, 0))
#> [,1] [,2] [,3]
#> [1,] 1 1 1
#> [2,] 0 0 0
```

*Guidotti, E. (2020). “calculus: High dimensional numerical and symbolic calculus in R”. https://arxiv.org/abs/2101.00086*

A BibTeX entry for LaTeX users is