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Turunan dari $$$\cos{\left(\ln\left(x\right) \right)}$$$
Kalkulator terkait: Kalkulator Diferensiasi Logaritmik, Kalkulator Diferensiasi Implisit dengan Langkah-langkah
Masukan Anda
Temukan $$$\frac{d}{dx} \left(\cos{\left(\ln\left(x\right) \right)}\right)$$$.
Solusi
Fungsi $$$\cos{\left(\ln\left(x\right) \right)}$$$ merupakan komposisi $$$f{\left(g{\left(x \right)} \right)}$$$ dari dua fungsi $$$f{\left(u \right)} = \cos{\left(u \right)}$$$ dan $$$g{\left(x \right)} = \ln\left(x\right)$$$.
Terapkan aturan rantai $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:
$${\color{red}\left(\frac{d}{dx} \left(\cos{\left(\ln\left(x\right) \right)}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(\cos{\left(u \right)}\right) \frac{d}{dx} \left(\ln\left(x\right)\right)\right)}$$Turunan fungsi kosinus adalah $$$\frac{d}{du} \left(\cos{\left(u \right)}\right) = - \sin{\left(u \right)}$$$:
$${\color{red}\left(\frac{d}{du} \left(\cos{\left(u \right)}\right)\right)} \frac{d}{dx} \left(\ln\left(x\right)\right) = {\color{red}\left(- \sin{\left(u \right)}\right)} \frac{d}{dx} \left(\ln\left(x\right)\right)$$Kembalikan ke variabel semula:
$$- \sin{\left({\color{red}\left(u\right)} \right)} \frac{d}{dx} \left(\ln\left(x\right)\right) = - \sin{\left({\color{red}\left(\ln\left(x\right)\right)} \right)} \frac{d}{dx} \left(\ln\left(x\right)\right)$$Turunan dari logaritma natural adalah $$$\frac{d}{dx} \left(\ln\left(x\right)\right) = \frac{1}{x}$$$:
$$- \sin{\left(\ln\left(x\right) \right)} {\color{red}\left(\frac{d}{dx} \left(\ln\left(x\right)\right)\right)} = - \sin{\left(\ln\left(x\right) \right)} {\color{red}\left(\frac{1}{x}\right)}$$Dengan demikian, $$$\frac{d}{dx} \left(\cos{\left(\ln\left(x\right) \right)}\right) = - \frac{\sin{\left(\ln\left(x\right) \right)}}{x}$$$.
Jawaban
$$$\frac{d}{dx} \left(\cos{\left(\ln\left(x\right) \right)}\right) = - \frac{\sin{\left(\ln\left(x\right) \right)}}{x}$$$A