Here I summarise the recursive rules for deriving (differentiating) compound expressions.

There are many different notations for differentiation. I will mainly use Lagrange’s here; although it is less precise than some others, it will make what follows concise.

So:

I will also write

,

*i.e.* composition of functions with a circle

,

*i.e.* multiplicative product of functions with a dot or juxtaposition;

these are both products but in different senses.

Note that mathematical notation can be a bit ambiguous or inconsistent. For example:

is the inverse function (compositionally), not the reciprocal (which is written ); but

is the (multiplicative) square of the result of the function.

### Fundamental Rules

This is a nearly minimal set of basic rules. Such rules are usually established from limit arguments on the definition of derivation.

[const] , (*i.e.* for constant with respect to *x*);

[id] ;

[+] ;

[×] ;

[o] ;

[^] ;

;

;

;

;

;

.

### More Rules

From these a little algebra may be used to obtain:

[~]

(from [const,+]: );

[1/]

(from [const,×]: );

[] ;

(from [const,o]: ).

These in turn lead to:

[-] ;

[÷] .

### Yet More Rules

These all lead to, for example:

;

;

;

.

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