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Pertanyaan

Buktikan bahwa turunan dari fungsi f left parenthesis x right parenthesis equals square root of 5 x plus 1 end root adalah 1 half open curly brackets open parentheses square root of 5 x plus 1 end root close parentheses to the power of negative 1 end exponent times 5 close curly brackets

A. Salim

Master Teacher

Mahasiswa/Alumni Universitas Pelita Harapan

Jawaban terverifikasi

Jawaban

terbukti bahwa turunan dari f left parenthesis x right parenthesis equals square root of 5 x plus 1 end root adalah 1 half open curly brackets open parentheses square root of 5 x plus 1 end root close parentheses to the power of negative 1 end exponent times 5 close curly brackets.

Pembahasan

Turunan dari f left parenthesis x right parenthesis equals square root of 5 x plus 1 end root dapat dicari dengan aturan rantai, yaitu jika y equals f left parenthesis u right parenthesis, di mana u merupakan fungsi g left parenthesis x right parenthesis, maka turunannya adalah fraction numerator d y over denominator d x end fraction equals fraction numerator d y over denominator d u end fraction cross times fraction numerator d u over denominator d x end fraction.


Dengan aturan rantai, turunan y equals f left parenthesis x right parenthesis equals square root of 5 x plus 1 end root adalah sebagai berikut:

Misal u equals 5 x plus 1, maka y equals u to the power of 1 half end exponent.

Jika u equals 5 x plus 1, maka fraction numerator d u over denominator d x end fraction equals 5.

Jika y equals u to the power of 1 half end exponent, maka fraction numerator d y over denominator d u end fraction equals 1 half u to the power of negative 1 half end exponent.

Maka, diperoleh:


table attributes columnalign right center left columnspacing 0px end attributes row cell f apostrophe left parenthesis x right parenthesis end cell equals cell fraction numerator d y over denominator d x end fraction equals fraction numerator d y over denominator d u end fraction cross times fraction numerator d u over denominator d x end fraction end cell row blank equals cell 1 half u to the power of negative 1 half end exponent cross times 5 end cell row blank equals cell 1 half open curly brackets open parentheses 5 x plus 1 close parentheses to the power of negative 1 half end exponent times 5 close curly brackets end cell row blank equals cell 1 half open curly brackets open parentheses square root of 5 x plus 1 end root close parentheses to the power of negative 1 end exponent times 5 close curly brackets end cell end table


Jadi, terbukti bahwa turunan dari f left parenthesis x right parenthesis equals square root of 5 x plus 1 end root adalah 1 half open curly brackets open parentheses square root of 5 x plus 1 end root close parentheses to the power of negative 1 end exponent times 5 close curly brackets.

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