Add derivative(ex, v) overload

Fix #684

Author: YingboMa

docs/src/highlevel.md

@@ -18,6 +18,7 @@ modelingtoolkitize
 ## Differentiation Functions
 
 ```@docs
+ModelingToolkit.derivative
 ModelingToolkit.gradient
 ModelingToolkit.jacobian
 ModelingToolkit.sparsejacobian

src/direct.jl

@@ -1,3 +1,19 @@
+"""
+```julia
+derivative(O, v; simplify = true)
+```
+
+A helper function for computing the derivative of an expression with respect to
+`var`.
+"""
+function derivative(O, v; simplify = true)
+    if O isa AbstractArray
+        Num[expand_derivatives(Differential(v)(value(o)), simplify) for o in O]
+    else
+        Num(expand_derivatives(Differential(v)(value(O)), simplify))
+    end
+end
+
 """
 ```julia
 gradient(O, vars::AbstractVector; simplify = true)
@@ -232,7 +248,9 @@ function toexpr(O; canonicalize=true)
     op = operation(O)
     args = arguments(O)
     if op isa Differential
-        return :(derivative($(toexpr(args[1]; canonicalize=canonicalize)),$(toexpr(op.x; canonicalize=canonicalize))))
+        ex = toexpr(args[1]; canonicalize=canonicalize)
+        wrt = toexpr(op.x; canonicalize=canonicalize)
+        return :(Differential($wrt)($ex))
     elseif op isa Sym
         isempty(args) && return nameof(op)
         return Expr(:call, toexpr(op; canonicalize=canonicalize), toexpr(args; canonicalize=canonicalize)...)

test/direct.jl

@@ -12,6 +12,11 @@ canonequal(a, b) = isequal(simplify(a), simplify(b))
     Differential(z)(Differential(y)(Differential(x)(t)))
 )
 
+@test canonequal(
+        ModelingToolkit.derivative(sin(cos(x)), x),
+        -sin(x) * cos(cos(x))
+       )
+
 eqs = [σ*(y-x),
        x*(ρ-z)-y,
        x*y - β*z]