The document discusses calculating partial derivatives of functions with two variables. It provides examples of calculating the partial derivatives of various functions, including ( ), ( ), ( ), and ( ). The solutions show applying the definition of partial derivatives to find ( ) and ( ). There are 11 examples in total that demonstrate calculating partial derivatives of different multi-variable functions.

1. numerosnamente 1
Derivadas de Funções reais de duas variáveis reais
1- Considere a função ( )
Calcule ( ) e ( )
Resolução:
( ) ( ) ( )
( ) ( ) ( )
2- Considere a função ( )
Calcule ( ) e ( )
Resolução:
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
3- Considere a função ( ) ( )
Calcule ( ) e ( )
Resolução:
( ) ( ) ; ( )
( ) ( ) ( )
4- Considere a função ( ) ( )
Calcule ( ) e ( )
Resolução:
( ) ( ) ( ) ( ( )) ( ) ( )
( ) ( ) ( )
( ) ( ) ( ) ( ( )) ( )
( ) ( )

2. numerosnamente 2
5- Considere a função ( ) ( )
Calcule ( ) e ( )
Resolução:
( ) ( ) ( ) ( ( )) ( ) ( )
( ) ( ) ( )
( ) ( ) ( ) ( ( )) ( )
( )
6- Considere a função ( )
Calcule e
Resolução:
( ) ( ) ( ) ;
( ) ( )( )
( ) ( ) ( )
( ) ( )
7- Considere a função ( ) ( )
Calcule e
Resolução:
( ) ( ) ( ) ( ( )) ( ) ( ( ))
(( ) ( ) ( ( )) ) (( ) ( ) ( ( )) )
( ) ( ) ( )
( ( ) ( ) ( ))
( ) ( ) ( ) ( ( )) ( )
( ) ( ( )) ( )(( ) ( ) ( ( )) )
( ( )) ( )(( ( ) ( ))
( ) ( ) ( ))

3. numerosnamente 3
( ( ) ( ) ( ))
=
8- Considere a função ( ) ( ) ( )
Calcule ( ) , ( ) , e
Resolução:
( ) ( ( )) ( ( )) ( ) ( )
( ) ( ( )) ( ( )) ( ) ( )
( ) ( ( )) ( ( )) ( ) ( )
( ) ( ( )) ( ( )) ( ) ( )
( ( )) ( ( ))
( ) ( ) ( )
( ( )) ( ( ))
( ) ( ) ( )
Note que
9- Considere a função ( ) ( )
Calcule e
Resolução:
( ) ( ) ( ( ) ( ( )) ( )
( )
( ) ( ( ) ( ( )) ( )
( ) ( ( ) )
( ) ( ) ( ( ) ( ( )) ( )
( )
( ) ( ( ) ( ( )) ( )

4. numerosnamente 4
( ) ( ( ) )
10- Considere a função ( ) {
( )
( )
Calcule ( ) e ( )
Resolução: Para funções definidas por ramos, aplica-se a definição de derivada de uma
função num ponto.
( )
( ) ( )
( )
( ) ( )
( )
( ) ( )
( )
( ) ( )
11- Considere a função ( ) ( ) ( )
Calcule ( ) e ( )
Resolução:
( ) ( ) ( )
( )
( ) ( ) ( )
( )