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Turunan dari $$$\sec^{3}{\left(u \right)}$$$
Kalkulator terkait: Kalkulator Diferensiasi Logaritmik, Kalkulator Diferensiasi Implisit dengan Langkah-langkah
Masukan Anda
Temukan $$$\frac{d}{du} \left(\sec^{3}{\left(u \right)}\right)$$$.
Solusi
Fungsi $$$\sec^{3}{\left(u \right)}$$$ merupakan komposisi $$$f{\left(g{\left(u \right)} \right)}$$$ dari dua fungsi $$$f{\left(v \right)} = v^{3}$$$ dan $$$g{\left(u \right)} = \sec{\left(u \right)}$$$.
Terapkan aturan rantai $$$\frac{d}{du} \left(f{\left(g{\left(u \right)} \right)}\right) = \frac{d}{dv} \left(f{\left(v \right)}\right) \frac{d}{du} \left(g{\left(u \right)}\right)$$$:
$${\color{red}\left(\frac{d}{du} \left(\sec^{3}{\left(u \right)}\right)\right)} = {\color{red}\left(\frac{d}{dv} \left(v^{3}\right) \frac{d}{du} \left(\sec{\left(u \right)}\right)\right)}$$Terapkan aturan pangkat $$$\frac{d}{dv} \left(v^{n}\right) = n v^{n - 1}$$$ dengan $$$n = 3$$$:
$${\color{red}\left(\frac{d}{dv} \left(v^{3}\right)\right)} \frac{d}{du} \left(\sec{\left(u \right)}\right) = {\color{red}\left(3 v^{2}\right)} \frac{d}{du} \left(\sec{\left(u \right)}\right)$$Kembalikan ke variabel semula:
$$3 {\color{red}\left(v\right)}^{2} \frac{d}{du} \left(\sec{\left(u \right)}\right) = 3 {\color{red}\left(\sec{\left(u \right)}\right)}^{2} \frac{d}{du} \left(\sec{\left(u \right)}\right)$$Turunan fungsi sekan adalah $$$\frac{d}{du} \left(\sec{\left(u \right)}\right) = \tan{\left(u \right)} \sec{\left(u \right)}$$$:
$$3 \sec^{2}{\left(u \right)} {\color{red}\left(\frac{d}{du} \left(\sec{\left(u \right)}\right)\right)} = 3 \sec^{2}{\left(u \right)} {\color{red}\left(\tan{\left(u \right)} \sec{\left(u \right)}\right)}$$Dengan demikian, $$$\frac{d}{du} \left(\sec^{3}{\left(u \right)}\right) = 3 \tan{\left(u \right)} \sec^{3}{\left(u \right)}$$$.
Jawaban
$$$\frac{d}{du} \left(\sec^{3}{\left(u \right)}\right) = 3 \tan{\left(u \right)} \sec^{3}{\left(u \right)}$$$A