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Turunan kedua dari $$$\ln\left(x\right)$$$
Kalkulator terkait: Kalkulator Turunan, Kalkulator Diferensiasi Logaritmik
Masukan Anda
Temukan $$$\frac{d^{2}}{dx^{2}} \left(\ln\left(x\right)\right)$$$.
Solusi
Tentukan turunan pertama $$$\frac{d}{dx} \left(\ln\left(x\right)\right)$$$
Turunan dari logaritma natural adalah $$$\frac{d}{dx} \left(\ln\left(x\right)\right) = \frac{1}{x}$$$:
$${\color{red}\left(\frac{d}{dx} \left(\ln\left(x\right)\right)\right)} = {\color{red}\left(\frac{1}{x}\right)}$$Dengan demikian, $$$\frac{d}{dx} \left(\ln\left(x\right)\right) = \frac{1}{x}$$$.
Selanjutnya, $$$\frac{d^{2}}{dx^{2}} \left(\ln\left(x\right)\right) = \frac{d}{dx} \left(\frac{1}{x}\right)$$$
Terapkan aturan pangkat $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ dengan $$$n = -1$$$:
$${\color{red}\left(\frac{d}{dx} \left(\frac{1}{x}\right)\right)} = {\color{red}\left(- \frac{1}{x^{2}}\right)}$$Dengan demikian, $$$\frac{d}{dx} \left(\frac{1}{x}\right) = - \frac{1}{x^{2}}$$$.
Oleh karena itu, $$$\frac{d^{2}}{dx^{2}} \left(\ln\left(x\right)\right) = - \frac{1}{x^{2}}$$$.
Jawaban
$$$\frac{d^{2}}{dx^{2}} \left(\ln\left(x\right)\right) = - \frac{1}{x^{2}}$$$A