Pertanyaan

Diketahui fungsi y = sec 2 x . Tentukan d x 2 d 2 y .

Diketahui fungsi $y = sec^{2} x$ . Tentukan $\frac{d ^{2} y}{d x ^{2}}$ .

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Jawaban terverifikasi

Jawaban

dari adalah .

$fraction numerator straight d squared y over denominator straight d x squared end fraction$ dari y equals sec squared x

adalah

6 sec to the power of 4 open parentheses x close parentheses minus 4 sec squared open parentheses x close parentheses

Pembahasan

Ingat bahwa: Mencari turunan pertama dari fungsi . Mencari turunan kedua dari fungsi : Jadi, dari adalah .

Ingat bahwa:

$fraction numerator straight d f left parenthesis u right parenthesis over denominator straight d x end fraction equals fraction numerator straight d f over denominator straight d u end fraction space times space fraction numerator straight d u over denominator straight d x end fraction fraction numerator straight d over denominator straight d x end fraction open parentheses sec space x close parentheses equals sec space x space tan space x fraction numerator straight d over denominator straight d x end fraction open parentheses tan space x close parentheses equals sec squared space x tan squared space x plus 1 equals sec squared space x$

Mencari turunan pertama dari fungsi y equals sec squared x .

$fraction numerator straight d over denominator straight d x end fraction open parentheses sec squared open parentheses x close parentheses close parentheses equals fraction numerator straight d over denominator straight d u end fraction open parentheses u squared close parentheses fraction numerator straight d over denominator straight d x end fraction open parentheses sec open parentheses x close parentheses close parentheses equals 2 u fraction numerator straight d over denominator straight d x end fraction open parentheses sec open parentheses x close parentheses close parentheses equals 2 sec open parentheses x close parentheses fraction numerator straight d over denominator straight d x end fraction open parentheses sec open parentheses x close parentheses close parentheses equals 2 sec open parentheses x close parentheses sec open parentheses x close parentheses tan open parentheses x close parentheses equals 2 sec squared open parentheses x close parentheses tan open parentheses x close parentheses$

Mencari turunan kedua dari fungsi y equals 2 sec squared open parentheses x close parentheses tan open parentheses x close parentheses :

$fraction numerator straight d over denominator straight d x end fraction open parentheses 2 sec squared open parentheses x close parentheses tan open parentheses x close parentheses close parentheses equals 2 fraction numerator straight d over denominator straight d x end fraction open parentheses sec squared open parentheses x close parentheses tan open parentheses x close parentheses close parentheses equals 2 open parentheses fraction numerator straight d over denominator straight d x end fraction open parentheses sec squared open parentheses x close parentheses close parentheses tan open parentheses x close parentheses plus fraction numerator straight d over denominator straight d x end fraction open parentheses tan open parentheses x close parentheses close parentheses sec squared open parentheses x close parentheses close parentheses equals 2 open parentheses 2 sec open parentheses x close parentheses fraction numerator straight d over denominator straight d x end fraction open parentheses sec open parentheses x close parentheses close parentheses plus sec squared open parentheses x close parentheses sec squared open parentheses x close parentheses close parentheses equals 2 open parentheses 2 sec open parentheses x close parentheses sec open parentheses x close parentheses tan open parentheses x close parentheses plus sec squared open parentheses x close parentheses sec squared open parentheses x close parentheses close parentheses equals 2 open parentheses 2 sec squared open parentheses x close parentheses tan open parentheses x close parentheses plus sec squared open parentheses x close parentheses sec squared open parentheses x close parentheses close parentheses equals 2 open parentheses sec to the power of 4 open parentheses x close parentheses plus 2 sec squared open parentheses x close parentheses tan squared open parentheses x close parentheses close parentheses equals 2 sec to the power of 4 open parentheses x close parentheses plus 4 sec squared open parentheses x close parentheses open parentheses sec squared open parentheses x close parentheses minus 1 close parentheses equals 2 sec to the power of 4 open parentheses x close parentheses plus 4 sec to the power of 4 open parentheses x close parentheses minus 4 sec squared open parentheses x close parentheses equals 6 sec to the power of 4 open parentheses x close parentheses minus 4 sec squared open parentheses x close parentheses$

Jadi, $fraction numerator straight d squared y over denominator straight d x squared end fraction$ dari y equals sec squared x adalah 6 sec to the power of 4 open parentheses x close parentheses minus 4 sec squared open parentheses x close parentheses .