Kalkulator Turunan Kedua
Hitung turunan kedua langkah demi langkah
Kalkulator ini akan menemukan turunan kedua dari fungsi apa pun, dengan langkah-langkah yang ditampilkan. Selain itu, kalkulator ini akan mengevaluasi turunan kedua pada titik yang diberikan jika diperlukan.
Kalkulator terkait: Kalkulator Turunan, Kalkulator Diferensiasi Logaritmik
Masukan Anda
Temukan $$$\frac{d^{2}}{dx^{2}} \left(\sin{\left(5 x \right)}\right)$$$.
Solusi
Tentukan turunan pertama $$$\frac{d}{dx} \left(\sin{\left(5 x \right)}\right)$$$
Fungsi $$$\sin{\left(5 x \right)}$$$ merupakan komposisi $$$f{\left(g{\left(x \right)} \right)}$$$ dari dua fungsi $$$f{\left(u \right)} = \sin{\left(u \right)}$$$ dan $$$g{\left(x \right)} = 5 x$$$.
Terapkan aturan rantai $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:
$${\color{red}\left(\frac{d}{dx} \left(\sin{\left(5 x \right)}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(\sin{\left(u \right)}\right) \frac{d}{dx} \left(5 x\right)\right)}$$Turunan fungsi sinus adalah $$$\frac{d}{du} \left(\sin{\left(u \right)}\right) = \cos{\left(u \right)}$$$:
$${\color{red}\left(\frac{d}{du} \left(\sin{\left(u \right)}\right)\right)} \frac{d}{dx} \left(5 x\right) = {\color{red}\left(\cos{\left(u \right)}\right)} \frac{d}{dx} \left(5 x\right)$$Kembalikan ke variabel semula:
$$\cos{\left({\color{red}\left(u\right)} \right)} \frac{d}{dx} \left(5 x\right) = \cos{\left({\color{red}\left(5 x\right)} \right)} \frac{d}{dx} \left(5 x\right)$$Terapkan aturan kelipatan konstanta $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ dengan $$$c = 5$$$ dan $$$f{\left(x \right)} = x$$$:
$$\cos{\left(5 x \right)} {\color{red}\left(\frac{d}{dx} \left(5 x\right)\right)} = \cos{\left(5 x \right)} {\color{red}\left(5 \frac{d}{dx} \left(x\right)\right)}$$Terapkan aturan pangkat $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ dengan $$$n = 1$$$, dengan kata lain, $$$\frac{d}{dx} \left(x\right) = 1$$$:
$$5 \cos{\left(5 x \right)} {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} = 5 \cos{\left(5 x \right)} {\color{red}\left(1\right)}$$Dengan demikian, $$$\frac{d}{dx} \left(\sin{\left(5 x \right)}\right) = 5 \cos{\left(5 x \right)}$$$.
Selanjutnya, $$$\frac{d^{2}}{dx^{2}} \left(\sin{\left(5 x \right)}\right) = \frac{d}{dx} \left(5 \cos{\left(5 x \right)}\right)$$$
Terapkan aturan kelipatan konstanta $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ dengan $$$c = 5$$$ dan $$$f{\left(x \right)} = \cos{\left(5 x \right)}$$$:
$${\color{red}\left(\frac{d}{dx} \left(5 \cos{\left(5 x \right)}\right)\right)} = {\color{red}\left(5 \frac{d}{dx} \left(\cos{\left(5 x \right)}\right)\right)}$$Fungsi $$$\cos{\left(5 x \right)}$$$ merupakan komposisi $$$f{\left(g{\left(x \right)} \right)}$$$ dari dua fungsi $$$f{\left(u \right)} = \cos{\left(u \right)}$$$ dan $$$g{\left(x \right)} = 5 x$$$.
Terapkan aturan rantai $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:
$$5 {\color{red}\left(\frac{d}{dx} \left(\cos{\left(5 x \right)}\right)\right)} = 5 {\color{red}\left(\frac{d}{du} \left(\cos{\left(u \right)}\right) \frac{d}{dx} \left(5 x\right)\right)}$$Turunan fungsi kosinus adalah $$$\frac{d}{du} \left(\cos{\left(u \right)}\right) = - \sin{\left(u \right)}$$$:
$$5 {\color{red}\left(\frac{d}{du} \left(\cos{\left(u \right)}\right)\right)} \frac{d}{dx} \left(5 x\right) = 5 {\color{red}\left(- \sin{\left(u \right)}\right)} \frac{d}{dx} \left(5 x\right)$$Kembalikan ke variabel semula:
$$- 5 \sin{\left({\color{red}\left(u\right)} \right)} \frac{d}{dx} \left(5 x\right) = - 5 \sin{\left({\color{red}\left(5 x\right)} \right)} \frac{d}{dx} \left(5 x\right)$$Terapkan aturan kelipatan konstanta $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ dengan $$$c = 5$$$ dan $$$f{\left(x \right)} = x$$$:
$$- 5 \sin{\left(5 x \right)} {\color{red}\left(\frac{d}{dx} \left(5 x\right)\right)} = - 5 \sin{\left(5 x \right)} {\color{red}\left(5 \frac{d}{dx} \left(x\right)\right)}$$Terapkan aturan pangkat $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ dengan $$$n = 1$$$, dengan kata lain, $$$\frac{d}{dx} \left(x\right) = 1$$$:
$$- 25 \sin{\left(5 x \right)} {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} = - 25 \sin{\left(5 x \right)} {\color{red}\left(1\right)}$$Dengan demikian, $$$\frac{d}{dx} \left(5 \cos{\left(5 x \right)}\right) = - 25 \sin{\left(5 x \right)}$$$.
Oleh karena itu, $$$\frac{d^{2}}{dx^{2}} \left(\sin{\left(5 x \right)}\right) = - 25 \sin{\left(5 x \right)}$$$.
Jawaban
$$$\frac{d^{2}}{dx^{2}} \left(\sin{\left(5 x \right)}\right) = - 25 \sin{\left(5 x \right)}$$$A