Simplify Derivative with Substitution

Question

I try to evaluate:

$$ \frac{\partial}{\partial x} \log{u(x, y, z)}$$

Mathematica gives:

$$ \frac{1}{x+y+z}$$

I want to simplify the expression with my function:

$$ \frac{1}{u(x, y, z)}$$

How to do that?

Thanks.

u[x_, y_, z_] = x + y + z
Simplify[D[Log[u[x, y, z]], x]]

kglr · Accepted Answer · 2018-11-22 18:05:37Z

7

D[Log[u[x, y, z]], x] /. u[x_, y_, z_] :> Defer[u[x, y, z]]

1/u[x, y, z]

answered Nov 22, 2018 at 17:36

kglr

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$\begingroup$ A more general substitution: /. u[x_,y_,z_] -> Defer[u[x,y,z]] $\endgroup$
– R zu
Commented Nov 22, 2018 at 18:04
$\begingroup$ @Rzu, good point. $\endgroup$
– kglr
Commented Nov 22, 2018 at 18:05

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Carl Woll · Accepted Answer · 2018-11-22 19:26:29Z

4

An alternative is to define UpValues instead of DownValues of u:

Derivative[1, 0, 0][u] ^:= 1&
Derivative[0, 1, 0][u] ^:= 1&
Derivative[0, 0, 1][u] ^:= 1&

D[Log[u[x, y, z]], x]

1/u[x, y, z]

answered Nov 22, 2018 at 19:26

Carl Woll

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$\begingroup$ What are UpValues and DownValues? The definition in the doc seems recursive: UpValue "gives a list of transformation rules corresponding to all upvalues defined for the symbol f. " $\endgroup$
– R zu
Commented Nov 22, 2018 at 19:29
$\begingroup$ @Rzu Maybe you can check out the documentation for UpSetDelayed and TagSetDelayed. $\endgroup$
– Carl Woll
Commented Nov 22, 2018 at 19:39

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2 Answers 2