# Exercici 10.17

## Apartat a)

kill(all)$

f(x,y,z):=x^4+3*x^2*z^2-4*x*y*z;

Derivades parcials de primer ordre

define (Dxf(x,y,z) , diff(f(x,y,z), x) );
define (Dyf(x,y,z) , diff(f(x,y,z), y) );
define (Dzf(x,y,z) , diff(f(x,y,z), z) );

Derivades parcials de segon ordre

define (Dxxf(x,y,z) , diff(Dxf(x,y,z), x) );
define (Dxyf(x,y,z) , diff(Dxf(x,y,z), y) );
define (Dxzf(x,y,z) , diff(Dxf(x,y,z), z) );

define (Dyxf(x,y,z) , diff(Dyf(x,y,z), x) );
define (Dyyf(x,y,z) , diff(Dyf(x,y,z), y) );
define (Dyzf(x,y,z) , diff(Dyf(x,y,z), z) );

define (Dzxf(x,y,z) , diff(Dzf(x,y,z), x) );
define (Dzyf(x,y,z) , diff(Dzf(x,y,z), y) );
define (Dzzf(x,y,z) , diff(Dzf(x,y,z), z) );

Derivades parcials de tercer ordre

define (Dxxxf(x,y,z) , diff(Dxxf(x,y,z), x) );
define (Dxxyf(x,y,z) , diff(Dxxf(x,y,z), y) );
define (Dxxzf(x,y,z) , diff(Dxxf(x,y,z), z) );

define (Dxyxf(x,y,z) , diff(Dxyf(x,y,z), x) );
define (Dxyyf(x,y,z) , diff(Dxyf(x,y,z), y) );
define (Dxyzf(x,y,z) , diff(Dxyf(x,y,z), z) );

define (Dxzxf(x,y,z) , diff(Dxzf(x,y,z), x) );
define (Dxzyf(x,y,z) , diff(Dxzf(x,y,z), y) );
define (Dxzzf(x,y,z) , diff(Dxzf(x,y,z), z) );

define (Dyxxf(x,y,z) , diff(Dyxf(x,y,z), x) );
define (Dyxyf(x,y,z) , diff(Dyxf(x,y,z), y) );
define (Dyxzf(x,y,z) , diff(Dyxf(x,y,z), z) );

define (Dyyxf(x,y,z) , diff(Dyyf(x,y,z), x) );
define (Dyyyf(x,y,z) , diff(Dyyf(x,y,z), y) );
define (Dyyzf(x,y,z) , diff(Dyyf(x,y,z), z) );

define (Dyzxf(x,y,z) , diff(Dyzf(x,y,z), x) );
define (Dyzyf(x,y,z) , diff(Dyzf(x,y,z), y) );
define (Dyzzf(x,y,z) , diff(Dyzf(x,y,z), z) );

define (Dzxxf(x,y,z) , diff(Dzxf(x,y,z), x) );
define (Dzxyf(x,y,z) , diff(Dzxf(x,y,z), y) );
define (Dzxzf(x,y,z) , diff(Dzxf(x,y,z), z) );

define (Dzyxf(x,y,z) , diff(Dzyf(x,y,z), x) );
define (Dzyyf(x,y,z) , diff(Dzyf(x,y,z), y) );
define (Dzyzf(x,y,z) , diff(Dzyf(x,y,z), z) );

define (Dzzxf(x,y,z) , diff(Dzzf(x,y,z), x) );
define (Dzzyf(x,y,z) , diff(Dzzf(x,y,z), y) );
define (Dzzzf(x,y,z) , diff(Dzzf(x,y,z), z) );

## Apartat b)

kill(all)$

f(x,y,z):=4*x*y*(%e^z)+x/(2*y)-sqrt(x^2*z+3*y^2);

Derivades parcials de primer ordre

define (Dxf(x,y,z) , diff(f(x,y,z), x) );
define (Dyf(x,y,z) , diff(f(x,y,z), y) );
define (Dzf(x,y,z) , diff(f(x,y,z), z) );

Derivades parcials de segon ordre

define (Dxxf(x,y,z) , diff(Dxf(x,y,z), x) );
define (Dxyf(x,y,z) , diff(Dxf(x,y,z), y) );
define (Dxzf(x,y,z) , diff(Dxf(x,y,z), z) );

define (Dyxf(x,y,z) , diff(Dyf(x,y,z), x) );
define (Dyyf(x,y,z) , diff(Dyf(x,y,z), y) );
define (Dyzf(x,y,z) , diff(Dyf(x,y,z), z) );

define (Dzxf(x,y,z) , diff(Dzf(x,y,z), x) );
define (Dzyf(x,y,z) , diff(Dzf(x,y,z), y) );
define (Dzzf(x,y,z) , diff(Dzf(x,y,z), z) );

Derivades parcials de tercer ordre

define (Dxxxf(x,y,z) , diff(Dxxf(x,y,z), x) );
define (Dxxyf(x,y,z) , diff(Dxxf(x,y,z), y) );
define (Dxxzf(x,y,z) , diff(Dxxf(x,y,z), z) );

define (Dxyxf(x,y,z) , diff(Dxyf(x,y,z), x) );
define (Dxyyf(x,y,z) , diff(Dxyf(x,y,z), y) );
define (Dxyzf(x,y,z) , diff(Dxyf(x,y,z), z) );

define (Dxzxf(x,y,z) , diff(Dxzf(x,y,z), x) );
define (Dxzyf(x,y,z) , diff(Dxzf(x,y,z), y) );
define (Dxzzf(x,y,z) , diff(Dxzf(x,y,z), z) );

define (Dyxxf(x,y,z) , diff(Dyxf(x,y,z), x) );
define (Dyxyf(x,y,z) , diff(Dyxf(x,y,z), y) );
define (Dyxzf(x,y,z) , diff(Dyxf(x,y,z), z) );

define (Dyyxf(x,y,z) , diff(Dyyf(x,y,z), x) );
define (Dyyyf(x,y,z) , diff(Dyyf(x,y,z), y) );
define (Dyyzf(x,y,z) , diff(Dyyf(x,y,z), z) );

define (Dyzxf(x,y,z) , diff(Dyzf(x,y,z), x) );
define (Dyzyf(x,y,z) , diff(Dyzf(x,y,z), y) );
define (Dyzzf(x,y,z) , diff(Dyzf(x,y,z), z) );

define (Dzxxf(x,y,z) , diff(Dzxf(x,y,z), x) );
define (Dzxyf(x,y,z) , diff(Dzxf(x,y,z), y) );
define (Dzxzf(x,y,z) , diff(Dzxf(x,y,z), z) );

define (Dzyxf(x,y,z) , diff(Dzyf(x,y,z), x) );
define (Dzyyf(x,y,z) , diff(Dzyf(x,y,z), y) );
define (Dzyzf(x,y,z) , diff(Dzyf(x,y,z), z) );

define (Dzzxf(x,y,z) , diff(Dzzf(x,y,z), x) );
define (Dzzyf(x,y,z) , diff(Dzzf(x,y,z), y) );
define (Dzzzf(x,y,z) , diff(Dzzf(x,y,z), z) );

## Apartat c)

kill(all)$

f(x,y,z):=exp(- y^2 / (2*x))+sin(x*y);

Derivades parcials de primer ordre

define (Dxf(x,y,z) , diff(f(x,y,z), x) );
define (Dyf(x,y,z) , diff(f(x,y,z), y) );
define (Dzf(x,y,z) , diff(f(x,y,z), z) );

Derivades parcials de segon ordre

define (Dxxf(x,y,z) , diff(Dxf(x,y,z), x) );
define (Dxyf(x,y,z) , diff(Dxf(x,y,z), y) );
define (Dxzf(x,y,z) , diff(Dxf(x,y,z), z) );

define (Dyxf(x,y,z) , diff(Dyf(x,y,z), x) );
define (Dyyf(x,y,z) , diff(Dyf(x,y,z), y) );
define (Dyzf(x,y,z) , diff(Dyf(x,y,z), z) );

define (Dzxf(x,y,z) , diff(Dzf(x,y,z), x) );
define (Dzyf(x,y,z) , diff(Dzf(x,y,z), y) );
define (Dzzf(x,y,z) , diff(Dzf(x,y,z), z) );

Derivades parcials de tercer ordre

define (Dxxxf(x,y,z) , diff(Dxxf(x,y,z), x) );
define (Dxxyf(x,y,z) , diff(Dxxf(x,y,z), y) );
define (Dxxzf(x,y,z) , diff(Dxxf(x,y,z), z) );

define (Dxyxf(x,y,z) , diff(Dxyf(x,y,z), x) );
define (Dxyyf(x,y,z) , diff(Dxyf(x,y,z), y) );
define (Dxyzf(x,y,z) , diff(Dxyf(x,y,z), z) );

define (Dxzxf(x,y,z) , diff(Dxzf(x,y,z), x) );
define (Dxzyf(x,y,z) , diff(Dxzf(x,y,z), y) );
define (Dxzzf(x,y,z) , diff(Dxzf(x,y,z), z) );

define (Dyxxf(x,y,z) , diff(Dyxf(x,y,z), x) );
define (Dyxyf(x,y,z) , diff(Dyxf(x,y,z), y) );
define (Dyxzf(x,y,z) , diff(Dyxf(x,y,z), z) );

define (Dyyxf(x,y,z) , diff(Dyyf(x,y,z), x) );
define (Dyyyf(x,y,z) , diff(Dyyf(x,y,z), y) );
define (Dyyzf(x,y,z) , diff(Dyyf(x,y,z), z) );

define (Dyzxf(x,y,z) , diff(Dyzf(x,y,z), x) );
define (Dyzyf(x,y,z) , diff(Dyzf(x,y,z), y) );
define (Dyzzf(x,y,z) , diff(Dyzf(x,y,z), z) );

define (Dzxxf(x,y,z) , diff(Dzxf(x,y,z), x) );
define (Dzxyf(x,y,z) , diff(Dzxf(x,y,z), y) );
define (Dzxzf(x,y,z) , diff(Dzxf(x,y,z), z) );

define (Dzyxf(x,y,z) , diff(Dzyf(x,y,z), x) );
define (Dzyyf(x,y,z) , diff(Dzyf(x,y,z), y) );
define (Dzyzf(x,y,z) , diff(Dzyf(x,y,z), z) );

define (Dzzxf(x,y,z) , diff(Dzzf(x,y,z), x) );
define (Dzzyf(x,y,z) , diff(Dzzf(x,y,z), y) );
define (Dzzzf(x,y,z) , diff(Dzzf(x,y,z), z) );

## Apartat d)

kill(all)$

f(x,y,z):=log((x+2*z^2) / (3*z*sqrt(y)));

Derivades parcials de primer ordre

define (Dxf(x,y,z) , diff(f(x,y,z), x) );
define (Dyf(x,y,z) , diff(f(x,y,z), y) );
define (Dzf(x,y,z) , diff(f(x,y,z), z) );

Derivades parcials de segon ordre

define (Dxxf(x,y,z) , diff(Dxf(x,y,z), x) );
define (Dxyf(x,y,z) , diff(Dxf(x,y,z), y) );
define (Dxzf(x,y,z) , diff(Dxf(x,y,z), z) );

define (Dyxf(x,y,z) , diff(Dyf(x,y,z), x) );
define (Dyyf(x,y,z) , diff(Dyf(x,y,z), y) );
define (Dyzf(x,y,z) , diff(Dyf(x,y,z), z) );

define (Dzxf(x,y,z) , diff(Dzf(x,y,z), x) );
define (Dzyf(x,y,z) , diff(Dzf(x,y,z), y) );
define (Dzzf(x,y,z) , diff(Dzf(x,y,z), z) );

Derivades parcials de tercer ordre

define (Dxxxf(x,y,z) , diff(Dxxf(x,y,z), x) );
define (Dxxyf(x,y,z) , diff(Dxxf(x,y,z), y) );
define (Dxxzf(x,y,z) , diff(Dxxf(x,y,z), z) );

define (Dxyxf(x,y,z) , diff(Dxyf(x,y,z), x) );
define (Dxyyf(x,y,z) , diff(Dxyf(x,y,z), y) );
define (Dxyzf(x,y,z) , diff(Dxyf(x,y,z), z) );

define (Dxzxf(x,y,z) , diff(Dxzf(x,y,z), x) );
define (Dxzyf(x,y,z) , diff(Dxzf(x,y,z), y) );
define (Dxzzf(x,y,z) , diff(Dxzf(x,y,z), z) );

define (Dyxxf(x,y,z) , diff(Dyxf(x,y,z), x) );
define (Dyxyf(x,y,z) , diff(Dyxf(x,y,z), y) );
define (Dyxzf(x,y,z) , diff(Dyxf(x,y,z), z) );

define (Dyyxf(x,y,z) , diff(Dyyf(x,y,z), x) );
define (Dyyyf(x,y,z) , diff(Dyyf(x,y,z), y) );
define (Dyyzf(x,y,z) , diff(Dyyf(x,y,z), z) );

define (Dyzxf(x,y,z) , diff(Dyzf(x,y,z), x) );
define (Dyzyf(x,y,z) , diff(Dyzf(x,y,z), y) );
define (Dyzzf(x,y,z) , diff(Dyzf(x,y,z), z) );

define (Dzxxf(x,y,z) , diff(Dzxf(x,y,z), x) );
define (Dzxyf(x,y,z) , diff(Dzxf(x,y,z), y) );
define (Dzxzf(x,y,z) , diff(Dzxf(x,y,z), z) );

define (Dzyxf(x,y,z) , diff(Dzyf(x,y,z), x) );
define (Dzyyf(x,y,z) , diff(Dzyf(x,y,z), y) );
define (Dzyzf(x,y,z) , diff(Dzyf(x,y,z), z) );

define (Dzzxf(x,y,z) , diff(Dzzf(x,y,z), x) );
define (Dzzyf(x,y,z) , diff(Dzzf(x,y,z), y) );
define (Dzzzf(x,y,z) , diff(Dzzf(x,y,z), z) );