# Derivative of a Symbolic Expression as a Symbol Valued Function

**URL:** <https://discourse.julialang.org/t/derivative-of-a-symbolic-expression-as-a-symbol-valued-function/60975>\
**Category:** New to Julia\
**Tags:** symbolics\
**Created:** [May 11, 2021, 7:01pm UTC](https://discourse.julialang.org/t/derivative-of-a-symbolic-expression-as-a-symbol-valued-function/60975 "2021-05-11T19:01:54Z")\
**Posts on this page:** 7\
**Page:** 1

<div class="post-metadata">

**Author:** ![GpuCoder](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/gpucoder/32/24511_2.png) [@GpuCoder](https://discourse.julialang.org/u/GpuCoder)\
**Post date:** [May 11, 2021, 7:01pm UTC](https://discourse.julialang.org/t/derivative-of-a-symbolic-expression-as-a-symbol-valued-function/60975/1 "2021-05-11T19:01:55Z")

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julia\> using Symbolics

```
julia> @variables a3, a2, a1, a0, t
5-element Vector{Num}:
 a3
 a2
 a1
 a0
 t

julia> f0(t) = a3*t^3 + a2*t^2 + a1*t + a0
f0 (generic function with 1 method)

julia> f1Gen = Differential(t)(f0(t))
Differential(t)(a0 + a1*t + a2*(t^2) + a3*(t^3))

julia> f1Exp = expand_derivatives(f1Gen)
a1 + 2a2*t + 3a3*(t^2)

julia> f1 = Symbolics.build_function(f1Exp, t)
:(function (t,)
      #= F:\Users\Redacted\.julia\packages\SymbolicUtils\9iQGH\src\code.jl:282 =#
      #= F:\Users\Redacted\.julia\packages\SymbolicUtils\9iQGH\src\code.jl:283 =#
      (+)(a1, (*)(2, a2, t), (*)(3, a3, (^)(t, 2)))
  end)

julia> eval(f1)(0)
a1

```

I wanted to take the symbolic derivative of a symbolic expression and then evaluate it at a numerical value of the argument producing a symbolic expression.

This solution seems to work, but violated my expectations (having written a lot of code for other CAS (Computer Algebra Systems)). It feels clunky, but I can imagine that with a little practice it might feel natural. Am I missing some shortcut?

The fact that

```
julia> f1Not(t) = expand_derivatives(f1Gen)
f1Not (generic function with 1 method)

```

produces something that looks right

```
julia> f1Not(t)
a1 + 2a2*t + 3a3*(t^2)

```

but is wrong

```
julia> f1Not(0)
a1 + 2a2*t + 3a3*(t^2)

```

created quite a bit of confusion for me.

The fact that `f0` is naturally the right type of expression may have increased my expectation that `expand_derivatives` could be induced to return the desired function directly.

```
julia> f0(0)
a0

```

My desire is to call some sort of “compile” step when creating a numerical valued function from a symbolic expression, because interpretation is too slow. But when producing a symbol valued function it seems that interpretation might be better … Maybe that’s incompatible with the Julia way …

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**Author:** ![ChrisRackauckas](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/chrisrackauckas/32/77_2.png) [@ChrisRackauckas](https://discourse.julialang.org/u/ChrisRackauckas)\
**Post date:** [May 11, 2021, 7:54pm UTC](https://discourse.julialang.org/t/derivative-of-a-symbolic-expression-as-a-symbol-valued-function/60975/2 "2021-05-11T19:54:09Z")

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Note that you could have done `Symbolics.derivative(f1Exp,t)`

> [@GpuCoder](#):
>
> My desire is to call some sort of “compile” step when creating a numerical valued function from a symbolic expression, because interpretation is too slow. But when producing a symbol valued function it seems that interpretation might be better … I don’t know …

Instead of creating a function you can just solve it by interpretation by using `substitute` and substituting in numerical values.

> [@GpuCoder](#):
>
> `f1Not(t) = expand_derivatives(f1Gen)`

You created a function that returns a constant.

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<div class="post-metadata">

**Author:** ![GpuCoder](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/gpucoder/32/24511_2.png) [@GpuCoder](https://discourse.julialang.org/u/GpuCoder)\
**Post date:** [May 11, 2021, 8:18pm UTC](https://discourse.julialang.org/t/derivative-of-a-symbolic-expression-as-a-symbol-valued-function/60975/3 "2021-05-11T20:18:42Z")

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> [@ChrisRackauckas](#):
>
> Instead of creating a function you can just solve it by interpretation by using `substitute` and substituting in numerical values.

This provided the desired symmetry to `build_funciton`.

The user knows which they want, and when … seems like a good system.

Didn’t catch on the first time around, because a type-o, led me to the wrong inference.

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<div class="post-metadata">

**Author:** ![ChrisRackauckas](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/chrisrackauckas/32/77_2.png) [@ChrisRackauckas](https://discourse.julialang.org/u/ChrisRackauckas)\
**Post date:** [May 11, 2021, 8:24pm UTC](https://discourse.julialang.org/t/derivative-of-a-symbolic-expression-as-a-symbol-valued-function/60975/4 "2021-05-11T20:24:51Z")

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No worries, we probably still need a bunch more work on the docs.

---

<div class="post-metadata">

**Author:** ![GpuCoder](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/gpucoder/32/24511_2.png) [@GpuCoder](https://discourse.julialang.org/u/GpuCoder)\
**Post date:** [May 11, 2021, 8:36pm UTC](https://discourse.julialang.org/t/derivative-of-a-symbolic-expression-as-a-symbol-valued-function/60975/5 "2021-05-11T20:36:03Z")

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- I’m not complaining, but yea, the documentation assumes too much context.
- I wish I could write some documentation for you, but I’m still figure things out.
- Maybe after I actually get some interesting amount of my own code to work.

---

<div class="post-metadata">

**Author:** ![ChrisRackauckas](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/chrisrackauckas/32/77_2.png) [@ChrisRackauckas](https://discourse.julialang.org/u/ChrisRackauckas)\
**Post date:** [May 11, 2021, 8:44pm UTC](https://discourse.julialang.org/t/derivative-of-a-symbolic-expression-as-a-symbol-valued-function/60975/6 "2021-05-11T20:44:06Z")

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You should complain about the docs, they need work 😆

---

<div class="post-metadata">

**Author:** ![GpuCoder](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/gpucoder/32/24511_2.png) [@GpuCoder](https://discourse.julialang.org/u/GpuCoder)\
**Post date:** [May 11, 2021, 8:55pm UTC](https://discourse.julialang.org/t/derivative-of-a-symbolic-expression-as-a-symbol-valued-function/60975/7 "2021-05-11T20:55:37Z")

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Perhaps. But I always fear the complaints won’t end up at “The Documentation Department”. 🤔