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Pertanyaan

Cari turunan parsial pertama fungsi yang diberikan. f ( x , y ) = ( 3 x 2 + y 2 ) − 3 1 ​

Cari turunan parsial pertama fungsi yang diberikan.

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G. Albiah

Master Teacher

Mahasiswa/Alumni Universitas Galuh Ciamis

Jawaban terverifikasi

Jawaban

turunan parsial pertama fungsi dari adalah dan.

turunan parsial pertama fungsi dari f left parenthesis x comma y right parenthesis equals open parentheses 3 x squared plus y squared close parentheses to the power of negative 1 third end exponent adalah  table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator d f over denominator d x end fraction end cell equals cell negative fraction numerator 2 x over denominator open parentheses 3 x squared plus y squared close parentheses to the power of 4 over 3 end exponent end fraction end cell end tabledan .table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator d f over denominator d y end fraction end cell equals cell negative fraction numerator 2 y over denominator 3 left parenthesis 3 x squared plus y squared right parenthesis to the power of 4 over 3 end exponent end fraction end cell end table

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Pembahasan

Misalkan : Masukan aturan berikut : Kemudian cari turunan pertama dari Untuk turunan , maka Maka turunan parsial adalah : Untuk turunan , maka Maka turunan parsial adalah : Jadi, turunan parsial pertama fungsi dari adalah dan.

Misalkan :

u equals 3 x squared plus y squared

Masukan aturan berikut :

table attributes columnalign right center left columnspacing 0px end attributes row cell u to the power of n end cell equals cell n times u to the power of n minus 1 end exponent end cell row cell u to the power of negative 1 third end exponent end cell equals cell negative 1 third u to the power of negative 1 third minus 1 end exponent end cell row blank equals cell negative 1 third u to the power of negative 1 third minus 3 over 3 end exponent end cell row blank equals cell negative 1 third u to the power of negative 4 over 3 end exponent end cell end table

Kemudian cari turunan pertama dari u equals open parentheses 3 x squared plus y squared close parentheses to the power of negative 1 third end exponent

Untuk turunan x, maka y equals 0

table attributes columnalign right center left columnspacing 0px end attributes row cell u to the power of n end cell equals cell n times u to the power of n minus 1 end exponent end cell row u equals cell open parentheses 3 x squared plus y squared close parentheses end cell row u equals cell 3 x squared plus 0 end cell row cell u apostrophe end cell equals cell 2 times 3 x to the power of 2 minus 1 end exponent end cell row cell u apostrophe end cell equals cell 6 x to the power of 1 end cell row cell u apostrophe end cell equals cell 6 x end cell end table

Maka turunan parsial  x adalah :

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator d f over denominator d x end fraction open parentheses 3 x squared plus y squared close parentheses to the power of 3 over 2 end exponent end cell equals cell u apostrophe cross times open parentheses negative 1 third u to the power of negative 4 over 3 end exponent close parentheses end cell row blank equals cell 6 x times negative 1 third open parentheses 3 x squared plus y squared close parentheses to the power of negative 4 over 3 end exponent end cell row blank equals cell negative fraction numerator 6 x over denominator 3 end fraction open parentheses 3 x squared plus y squared close parentheses to the power of negative 4 over 3 end exponent end cell row blank equals cell negative fraction numerator 2 x over denominator open parentheses 3 x squared plus y squared close parentheses to the power of 4 over 3 end exponent end fraction end cell end table

Untuk turunan y, maka x equals 0

table attributes columnalign right center left columnspacing 0px end attributes row cell u to the power of n end cell equals cell n times u to the power of n minus 1 end exponent end cell row u equals cell 3 x squared plus y squared end cell row u equals cell 0 plus y squared end cell row u equals cell y squared end cell row cell u apostrophe end cell equals cell 2 times open parentheses 1 close parentheses y to the power of 2 minus 1 end exponent end cell row cell u apostrophe end cell equals cell 2 cross times y to the power of 1 end cell row cell u apostrophe end cell equals cell 2 y to the power of 1 end cell row cell u apostrophe end cell equals cell 2 y end cell end table

Maka turunan parsial  y adalah :

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator d f over denominator d y end fraction left parenthesis 4 x minus y squared right parenthesis to the power of 3 over 2 end exponent end cell equals cell u apostrophe cross times open parentheses negative 1 third u to the power of negative 4 over 3 end exponent close parentheses end cell row blank equals cell 2 y cross times open parentheses negative 1 third left parenthesis 3 x squared plus y squared right parenthesis close parentheses to the power of negative 4 over 3 end exponent end cell row blank equals cell fraction numerator negative 2 y over denominator 3 end fraction left parenthesis 3 x squared plus y squared right parenthesis to the power of negative 4 over 3 end exponent end cell row blank equals cell negative fraction numerator 2 y over denominator 3 left parenthesis 3 x squared plus y squared right parenthesis to the power of 4 over 3 end exponent end fraction end cell end table

Jadi, turunan parsial pertama fungsi dari f left parenthesis x comma y right parenthesis equals open parentheses 3 x squared plus y squared close parentheses to the power of negative 1 third end exponent adalah  table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator d f over denominator d x end fraction end cell equals cell negative fraction numerator 2 x over denominator open parentheses 3 x squared plus y squared close parentheses to the power of 4 over 3 end exponent end fraction end cell end tabledan .table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator d f over denominator d y end fraction end cell equals cell negative fraction numerator 2 y over denominator 3 left parenthesis 3 x squared plus y squared right parenthesis to the power of 4 over 3 end exponent end fraction end cell end table

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