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1. Diketahui . Tunjukkan bahwa

Pertanyaan

1. Diketahui y equals fraction numerator cos space x over denominator 1 minus sin space x end fraction. Tunjukkan bahwa cos space x times fraction numerator d y over denominator d x end fraction equals y 

Pembahasan Soal:

Turunan fungsi f open parentheses x close parentheses equals sin space g open parentheses x close parentheses yaitu f apostrophe open parentheses x close parentheses equals cos space g open parentheses x close parentheses space g apostrophe open parentheses x close parentheses, dan untuk f open parentheses x close parentheses equals cos space g open parentheses x close parentheses maka turunannya yaitu f apostrophe open parentheses x close parentheses equals negative sin space g open parentheses x close parentheses space g apostrophe open parentheses x close parentheses.

Aturan pembagian pada turunan yaitu

 f open parentheses x close parentheses equals u over v f apostrophe open parentheses x close parentheses equals fraction numerator u apostrophe v minus u v apostrophe over denominator v squared end fraction 

Untuk menyelesaikan soal di atas, perhatikan penghitungan berikut:

    y equals fraction numerator cos space x over denominator 1 minus sin space x end fraction u equals cos space x rightwards arrow u apostrophe equals negative sin space x v equals 1 minus sin space x rightwards arrow v apostrophe equals 0 minus cos space x equals negative cos space x fraction numerator d y over denominator d x end fraction equals fraction numerator u apostrophe v minus u v apostrophe over denominator v squared end fraction fraction numerator d y over denominator d x end fraction equals fraction numerator open parentheses negative sin space x close parentheses open parentheses 1 minus sin space x close parentheses minus cos space x open parentheses negative cos space x close parentheses over denominator open parentheses 1 minus sin space x close parentheses squared end fraction fraction numerator d y over denominator d x end fraction equals fraction numerator negative sin space x plus sin squared space x plus cos squared space x over denominator open parentheses 1 minus sin space x close parentheses squared end fraction fraction numerator d y over denominator d x end fraction equals fraction numerator negative sin space x plus 1 over denominator open parentheses 1 minus sin space x close parentheses squared end fraction fraction numerator d y over denominator d x end fraction equals fraction numerator 1 minus sin space x over denominator open parentheses 1 minus sin space x close parentheses squared end fraction fraction numerator d y over denominator d x end fraction equals fraction numerator 1 over denominator 1 minus sin space x end fraction cos space x times fraction numerator d y over denominator d x end fraction equals cos space x times fraction numerator 1 over denominator 1 minus sin space x end fraction cos space x times fraction numerator d y over denominator d x end fraction equals fraction numerator cos space x over denominator 1 minus sin space x end fraction cos space x times fraction numerator d y over denominator d x end fraction equals y   

Jadi, terbukti bahwa cos space x times fraction numerator d y over denominator d x end fraction equals yspace 

Pembahasan terverifikasi oleh Roboguru

Terakhir diupdate 19 Juli 2021

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