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Tentukan turunan pertama fungsi-fungsi di bawah ini. c.

Tentukan turunan pertama begin mathsize 14px style open parentheses fraction numerator d y over denominator d x end fraction close parentheses end style fungsi-fungsi di bawah ini.

c. begin mathsize 14px style x equals square root of 4 minus 9 y squared end root end style 

Jawaban:

Ingat rumus aturan rantai turunan adalah sebagai berikut

begin mathsize 14px style fraction numerator d y over denominator d x end fraction equals fraction numerator d y over denominator d u end fraction times fraction numerator d u over denominator d x end fraction end style 

Diketahui begin mathsize 14px style x equals square root of 4 minus 9 y squared end root end style, maka fungsi begin mathsize 14px style y end style diperoleh sebagai berikut.

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row x equals cell square root of 4 minus 9 y squared end root end cell row cell square root of 4 minus 9 y squared end root end cell equals x row cell 4 minus 9 y squared end cell equals cell x squared end cell row cell negative 9 y squared end cell equals cell x squared minus 4 end cell row cell y squared end cell equals cell fraction numerator x squared minus 4 over denominator negative 9 end fraction end cell row y equals cell square root of fraction numerator x squared minus 4 over denominator negative 9 end fraction end root end cell end table end style 

Misalkan begin mathsize 14px style u equals fraction numerator x squared minus 4 over denominator negative 9 end fraction end style, maka diperoleh turunan pertama begin mathsize 14px style u end style sebagai berikut.

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row u equals cell fraction numerator x squared minus 4 over denominator negative 9 end fraction end cell row u equals cell negative 1 over 9 open parentheses x squared minus 4 close parentheses end cell row cell fraction numerator d u over denominator d x end fraction end cell equals cell negative 1 over 9 open parentheses 2 x close parentheses end cell row blank equals cell fraction numerator negative 2 x over denominator 9 end fraction end cell end table end style 

Dengan menggunakan aturan rantai, maka diperoleh turunan pertama begin mathsize 14px style fraction numerator d y over denominator d x end fraction end style dari undefined sebagai berikut.

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row x equals cell square root of 4 minus 9 y squared end root end cell row y equals cell square root of fraction numerator x squared minus 4 over denominator negative 9 end fraction end root end cell row cell fraction numerator d y over denominator d x end fraction end cell equals cell fraction numerator d y over denominator d u end fraction times fraction numerator d u over denominator d x end fraction end cell row blank equals cell fraction numerator d open parentheses square root of u close parentheses over denominator d u end fraction. fraction numerator negative 2 x over denominator 9 end fraction end cell row blank equals cell fraction numerator 1 over denominator 2 square root of u end fraction times fraction numerator negative 2 x over denominator 9 end fraction end cell row blank equals cell fraction numerator 1 over denominator square root of begin display style fraction numerator x squared minus 4 over denominator negative 9 end fraction end style end root end fraction. fraction numerator negative x over denominator 9 end fraction end cell row blank equals cell fraction numerator 1 over denominator begin display style 1 third end style square root of negative x squared plus 4 end root end fraction times fraction numerator negative x over denominator 9 end fraction end cell row blank equals cell fraction numerator 3 over denominator square root of negative x squared plus 4 end root end fraction times fraction numerator negative x over denominator 9 end fraction end cell row blank equals cell fraction numerator negative x over denominator 3 square root of negative x squared plus 4 end root end fraction end cell end table end style 

Dengan demikian turunan pertamanya adalah begin mathsize 14px style fraction numerator negative x over denominator 3 square root of negative x squared plus 4 end root end fraction end style.

 

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