# ForwardDiff.jl

Forward Mode Automatic Differentiation for Julia

Go To ForwardDiff’s Documentation

# ForwardDiff.jl

ForwardDiff implements methods to take derivatives, gradients, Jacobians, Hessians, and higher-order derivatives of native Julia functions (or any callable object, really) using forward mode automatic differentiation (AD).

While performance can vary depending on the functions you evaluate, the algorithms implemented by ForwardDiff generally outperform non-AD algorithms in both speed and accuracy.

Here’s a simple example showing the package in action:

``````julia> using ForwardDiff

julia> f(x::Vector) = sum(sin, x) + prod(tan, x) * sum(sqrt, x);

julia> x = rand(5) # small size for example's sake
5-element Array{Float64,1}:
0.986403
0.140913
0.294963
0.837125
0.650451

julia> g = x -> ForwardDiff.gradient(f, x); # g = ∇f

julia> g(x)
5-element Array{Float64,1}:
1.01358
2.50014
1.72574
1.10139
1.2445

julia> ForwardDiff.hessian(f, x)
5x5 Array{Float64,2}:
0.585111  3.48083  1.7706    0.994057  1.03257
3.48083   1.06079  5.79299   3.25245   3.37871
1.7706    5.79299  0.423981  1.65416   1.71818
0.994057  3.25245  1.65416   0.251396  0.964566
1.03257   3.37871  1.71818   0.964566  0.140689
``````

## Publications

If you find ForwardDiff useful in your work, we kindly request that you cite the following paper:

``````@article{RevelsLubinPapamarkou2016,
title = {Forward-Mode Automatic Differentiation in Julia},
author = {{Revels}, J. and {Lubin}, M. and {Papamarkou}, T.},
journal = {arXiv:1607.07892 [cs.MS]},
year = {2016},
url = {https://arxiv.org/abs/1607.07892}
}
``````