Update README.md

Author: RafaelArutjunjan

README.md

@@ -5,4 +5,43 @@
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-This package provides a front-end for differentiation operations in Julia that allows for code which is agnostic with respect to many of the available automatic and symbolic differentiation tools available in Julia. Moreover, the differentiation operators provided by **DerivableFunctions.jl** are also overloaded to allow for passthrough of symbolic variables. That is, if symbolic types such as `Symbolics.Num` are detected, the differentiation functors automatically switch to symbolic differentiation.
+This package provides a front-end for differentiation operations in Julia that allows for code which is agnostic with respect to many of the available automatic and symbolic differentiation tools available in Julia. Moreover, the differentiation operators provided by **DerivableFunctions.jl** are also overloaded to allow for passthrough of symbolic variables. That is, if symbolic types such as `Symbolics.Num` are detected, the differentiation operators automatically switch to symbolic differentiation.
+
+```julia
+julia> D = DFunction(x->[exp(x[1]^2 - x[2]), log(sin(x[2]))])
+(::DerivableFunction) (generic function with 1 method)
+
+julia> EvalF(D,[1,2])
+2-element Vector{Float64}:
+  0.36787944117144233
+ -0.09508303609516061
+
+julia> EvaldF(D,[1,2])
+2×2 Matrix{Float64}:
+ 0.735759  -0.367879
+ 0.0       -0.457658
+
+julia> EvalddF(D,[1,2])
+2×2×2 Array{Float64, 3}:
+[:, :, 1] =
+ 2.20728  -0.735759
+ 0.0       0.0
+
+[:, :, 2] =
+ -0.735759   0.367879
+  0.0       -1.20945
+
+julia> using Symbolics; @variables z[1:2]
+1-element Vector{Symbolics.Arr{Num, 1}}:
+ z[1:2]
+
+julia> EvalddF(D, z)
+2×2×2 Array{Num, 3}:
+[:, :, 1] =
+ 2exp(z[1]^2 - z[2]) + 4(z[1]^2)*exp(z[1]^2 - z[2])  -2exp(z[1]^2 - z[2])*z[1]
+ 0                                                    0
+
+[:, :, 2] =
+ -2exp(z[1]^2 - z[2])*z[1]                                                             exp(z[1]^2 - z[2])
+  0                         (-(cos(z[2])^2)) / (sin(z[2])^2) + (-sin(z[2])) / sin(z[2])
+```