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Turunan kedua dari $$$\csc{\left(x \right)}$$$

Kalkulator akan menghitung turunan kedua dari $$$\csc{\left(x \right)}$$$, beserta langkah-langkahnya.

Kalkulator terkait: Kalkulator Turunan, Kalkulator Diferensiasi Logaritmik

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Masukan Anda

Temukan $$$\frac{d^{2}}{dx^{2}} \left(\csc{\left(x \right)}\right)$$$.

Solusi

Tentukan turunan pertama $$$\frac{d}{dx} \left(\csc{\left(x \right)}\right)$$$

Turunan dari kosekan adalah $$$\frac{d}{dx} \left(\csc{\left(x \right)}\right) = - \cot{\left(x \right)} \csc{\left(x \right)}$$$:

$${\color{red}\left(\frac{d}{dx} \left(\csc{\left(x \right)}\right)\right)} = {\color{red}\left(- \cot{\left(x \right)} \csc{\left(x \right)}\right)}$$

Dengan demikian, $$$\frac{d}{dx} \left(\csc{\left(x \right)}\right) = - \cot{\left(x \right)} \csc{\left(x \right)}$$$.

Selanjutnya, $$$\frac{d^{2}}{dx^{2}} \left(\csc{\left(x \right)}\right) = \frac{d}{dx} \left(- \cot{\left(x \right)} \csc{\left(x \right)}\right)$$$

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 $$$c = -1$$$ dan $$$f{\left(x \right)} = \cot{\left(x \right)} \csc{\left(x \right)}$$$:

$${\color{red}\left(\frac{d}{dx} \left(- \cot{\left(x \right)} \csc{\left(x \right)}\right)\right)} = {\color{red}\left(- \frac{d}{dx} \left(\cot{\left(x \right)} \csc{\left(x \right)}\right)\right)}$$

Terapkan aturan hasil kali $$$\frac{d}{dx} \left(f{\left(x \right)} g{\left(x \right)}\right) = \frac{d}{dx} \left(f{\left(x \right)}\right) g{\left(x \right)} + f{\left(x \right)} \frac{d}{dx} \left(g{\left(x \right)}\right)$$$ pada $$$f{\left(x \right)} = \cot{\left(x \right)}$$$ dan $$$g{\left(x \right)} = \csc{\left(x \right)}$$$:

$$- {\color{red}\left(\frac{d}{dx} \left(\cot{\left(x \right)} \csc{\left(x \right)}\right)\right)} = - {\color{red}\left(\frac{d}{dx} \left(\cot{\left(x \right)}\right) \csc{\left(x \right)} + \cot{\left(x \right)} \frac{d}{dx} \left(\csc{\left(x \right)}\right)\right)}$$

Turunan dari kosekan adalah $$$\frac{d}{dx} \left(\csc{\left(x \right)}\right) = - \cot{\left(x \right)} \csc{\left(x \right)}$$$:

$$- \cot{\left(x \right)} {\color{red}\left(\frac{d}{dx} \left(\csc{\left(x \right)}\right)\right)} - \csc{\left(x \right)} \frac{d}{dx} \left(\cot{\left(x \right)}\right) = - \cot{\left(x \right)} {\color{red}\left(- \cot{\left(x \right)} \csc{\left(x \right)}\right)} - \csc{\left(x \right)} \frac{d}{dx} \left(\cot{\left(x \right)}\right)$$

Turunan dari kotangen adalah $$$\frac{d}{dx} \left(\cot{\left(x \right)}\right) = - \csc^{2}{\left(x \right)}$$$:

$$\cot^{2}{\left(x \right)} \csc{\left(x \right)} - \csc{\left(x \right)} {\color{red}\left(\frac{d}{dx} \left(\cot{\left(x \right)}\right)\right)} = \cot^{2}{\left(x \right)} \csc{\left(x \right)} - \csc{\left(x \right)} {\color{red}\left(- \csc^{2}{\left(x \right)}\right)}$$

Sederhanakan:

$$\cot^{2}{\left(x \right)} \csc{\left(x \right)} + \csc^{3}{\left(x \right)} = \left(-1 + \frac{2}{\sin^{2}{\left(x \right)}}\right) \csc{\left(x \right)}$$

Dengan demikian, $$$\frac{d}{dx} \left(- \cot{\left(x \right)} \csc{\left(x \right)}\right) = \left(-1 + \frac{2}{\sin^{2}{\left(x \right)}}\right) \csc{\left(x \right)}$$$.

Oleh karena itu, $$$\frac{d^{2}}{dx^{2}} \left(\csc{\left(x \right)}\right) = \left(-1 + \frac{2}{\sin^{2}{\left(x \right)}}\right) \csc{\left(x \right)}$$$.

Jawaban

$$$\frac{d^{2}}{dx^{2}} \left(\csc{\left(x \right)}\right) = \left(-1 + \frac{2}{\sin^{2}{\left(x \right)}}\right) \csc{\left(x \right)}$$$A