Dérivée de $$$\ln\left(x\right) \sin{\left(9 x \right)}$$$

Soit $$$H{\left(x \right)} = \ln\left(x\right) \sin{\left(9 x \right)}$$$.

Prenez le logarithme des deux membres : $$$\ln\left(H{\left(x \right)}\right) = \ln\left(\ln\left(x\right) \sin{\left(9 x \right)}\right)$$$.

Réécrivez le second membre en utilisant les propriétés des logarithmes : $$$\ln\left(H{\left(x \right)}\right) = \ln\left(\ln\left(x\right)\right) + \ln\left(\sin{\left(9 x \right)}\right)$$$

Dérivez séparément les deux membres de l’équation : $$$\frac{d}{dx} \left(\ln\left(H{\left(x \right)}\right)\right) = \frac{d}{dx} \left(\ln\left(\ln\left(x\right)\right) + \ln\left(\sin{\left(9 x \right)}\right)\right)$$$.

Dérivez le membre gauche de l’équation.

La fonction $$$\ln\left(H{\left(x \right)}\right)$$$ est la composée $$$f{\left(g{\left(x \right)} \right)}$$$ de deux fonctions $$$f{\left(u \right)} = \ln\left(u\right)$$$ et $$$g{\left(x \right)} = H{\left(x \right)}$$$.

Appliquez la règle de la chaîne $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:

$${\color{red}\left(\frac{d}{dx} \left(\ln\left(H{\left(x \right)}\right)\right)\right)} = {\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right) \frac{d}{dx} \left(H{\left(x \right)}\right)\right)}$$

La dérivée du logarithme naturel est $$$\frac{d}{du} \left(\ln\left(u\right)\right) = \frac{1}{u}$$$ :

$${\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right)\right)} \frac{d}{dx} \left(H{\left(x \right)}\right) = {\color{red}\left(\frac{1}{u}\right)} \frac{d}{dx} \left(H{\left(x \right)}\right)$$

Revenir à la variable initiale:

$$\frac{\frac{d}{dx} \left(H{\left(x \right)}\right)}{{\color{red}\left(u\right)}} = \frac{\frac{d}{dx} \left(H{\left(x \right)}\right)}{{\color{red}\left(H{\left(x \right)}\right)}}$$

Ainsi, $$$\frac{d}{dx} \left(\ln\left(H{\left(x \right)}\right)\right) = \frac{\frac{d}{dx} \left(H{\left(x \right)}\right)}{H{\left(x \right)}}$$$.

Dérivez le membre de droite de l’équation.

La dérivée d'une somme/différence est la somme/différence des dérivées :

$${\color{red}\left(\frac{d}{dx} \left(\ln\left(\ln\left(x\right)\right) + \ln\left(\sin{\left(9 x \right)}\right)\right)\right)} = {\color{red}\left(\frac{d}{dx} \left(\ln\left(\ln\left(x\right)\right)\right) + \frac{d}{dx} \left(\ln\left(\sin{\left(9 x \right)}\right)\right)\right)}$$

La fonction $$$\ln\left(\ln\left(x\right)\right)$$$ est la composée $$$f{\left(g{\left(x \right)} \right)}$$$ de deux fonctions $$$f{\left(u \right)} = \ln\left(u\right)$$$ et $$$g{\left(x \right)} = \ln\left(x\right)$$$.

Appliquez la règle de la chaîne $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:

$${\color{red}\left(\frac{d}{dx} \left(\ln\left(\ln\left(x\right)\right)\right)\right)} + \frac{d}{dx} \left(\ln\left(\sin{\left(9 x \right)}\right)\right) = {\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right) \frac{d}{dx} \left(\ln\left(x\right)\right)\right)} + \frac{d}{dx} \left(\ln\left(\sin{\left(9 x \right)}\right)\right)$$

La dérivée du logarithme naturel est $$$\frac{d}{du} \left(\ln\left(u\right)\right) = \frac{1}{u}$$$ :

$${\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right)\right)} \frac{d}{dx} \left(\ln\left(x\right)\right) + \frac{d}{dx} \left(\ln\left(\sin{\left(9 x \right)}\right)\right) = {\color{red}\left(\frac{1}{u}\right)} \frac{d}{dx} \left(\ln\left(x\right)\right) + \frac{d}{dx} \left(\ln\left(\sin{\left(9 x \right)}\right)\right)$$

Revenir à la variable initiale:

$$\frac{d}{dx} \left(\ln\left(\sin{\left(9 x \right)}\right)\right) + \frac{\frac{d}{dx} \left(\ln\left(x\right)\right)}{{\color{red}\left(u\right)}} = \frac{d}{dx} \left(\ln\left(\sin{\left(9 x \right)}\right)\right) + \frac{\frac{d}{dx} \left(\ln\left(x\right)\right)}{{\color{red}\left(\ln\left(x\right)\right)}}$$

La fonction $$$\ln\left(\sin{\left(9 x \right)}\right)$$$ est la composée $$$f{\left(g{\left(x \right)} \right)}$$$ de deux fonctions $$$f{\left(u \right)} = \ln\left(u\right)$$$ et $$$g{\left(x \right)} = \sin{\left(9 x \right)}$$$.

Appliquez la règle de la chaîne $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:

$${\color{red}\left(\frac{d}{dx} \left(\ln\left(\sin{\left(9 x \right)}\right)\right)\right)} + \frac{\frac{d}{dx} \left(\ln\left(x\right)\right)}{\ln\left(x\right)} = {\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right) \frac{d}{dx} \left(\sin{\left(9 x \right)}\right)\right)} + \frac{\frac{d}{dx} \left(\ln\left(x\right)\right)}{\ln\left(x\right)}$$

La dérivée du logarithme naturel est $$$\frac{d}{du} \left(\ln\left(u\right)\right) = \frac{1}{u}$$$ :

$${\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right)\right)} \frac{d}{dx} \left(\sin{\left(9 x \right)}\right) + \frac{\frac{d}{dx} \left(\ln\left(x\right)\right)}{\ln\left(x\right)} = {\color{red}\left(\frac{1}{u}\right)} \frac{d}{dx} \left(\sin{\left(9 x \right)}\right) + \frac{\frac{d}{dx} \left(\ln\left(x\right)\right)}{\ln\left(x\right)}$$

Revenir à la variable initiale:

$$\frac{\frac{d}{dx} \left(\sin{\left(9 x \right)}\right)}{{\color{red}\left(u\right)}} + \frac{\frac{d}{dx} \left(\ln\left(x\right)\right)}{\ln\left(x\right)} = \frac{\frac{d}{dx} \left(\sin{\left(9 x \right)}\right)}{{\color{red}\left(\sin{\left(9 x \right)}\right)}} + \frac{\frac{d}{dx} \left(\ln\left(x\right)\right)}{\ln\left(x\right)}$$

La fonction $$$\sin{\left(9 x \right)}$$$ est la composée $$$f{\left(g{\left(x \right)} \right)}$$$ de deux fonctions $$$f{\left(u \right)} = \sin{\left(u \right)}$$$ et $$$g{\left(x \right)} = 9 x$$$.

Appliquez la règle de la chaîne $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:

$$\frac{{\color{red}\left(\frac{d}{dx} \left(\sin{\left(9 x \right)}\right)\right)}}{\sin{\left(9 x \right)}} + \frac{\frac{d}{dx} \left(\ln\left(x\right)\right)}{\ln\left(x\right)} = \frac{{\color{red}\left(\frac{d}{du} \left(\sin{\left(u \right)}\right) \frac{d}{dx} \left(9 x\right)\right)}}{\sin{\left(9 x \right)}} + \frac{\frac{d}{dx} \left(\ln\left(x\right)\right)}{\ln\left(x\right)}$$

La dérivée du sinus est $$$\frac{d}{du} \left(\sin{\left(u \right)}\right) = \cos{\left(u \right)}$$$ :

$$\frac{{\color{red}\left(\frac{d}{du} \left(\sin{\left(u \right)}\right)\right)} \frac{d}{dx} \left(9 x\right)}{\sin{\left(9 x \right)}} + \frac{\frac{d}{dx} \left(\ln\left(x\right)\right)}{\ln\left(x\right)} = \frac{{\color{red}\left(\cos{\left(u \right)}\right)} \frac{d}{dx} \left(9 x\right)}{\sin{\left(9 x \right)}} + \frac{\frac{d}{dx} \left(\ln\left(x\right)\right)}{\ln\left(x\right)}$$

Revenir à la variable initiale:

$$\frac{\cos{\left({\color{red}\left(u\right)} \right)} \frac{d}{dx} \left(9 x\right)}{\sin{\left(9 x \right)}} + \frac{\frac{d}{dx} \left(\ln\left(x\right)\right)}{\ln\left(x\right)} = \frac{\cos{\left({\color{red}\left(9 x\right)} \right)} \frac{d}{dx} \left(9 x\right)}{\sin{\left(9 x \right)}} + \frac{\frac{d}{dx} \left(\ln\left(x\right)\right)}{\ln\left(x\right)}$$

Appliquez la règle du facteur constant $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ avec $$$c = 9$$$ et $$$f{\left(x \right)} = x$$$:

$$\frac{\cos{\left(9 x \right)} {\color{red}\left(\frac{d}{dx} \left(9 x\right)\right)}}{\sin{\left(9 x \right)}} + \frac{\frac{d}{dx} \left(\ln\left(x\right)\right)}{\ln\left(x\right)} = \frac{\cos{\left(9 x \right)} {\color{red}\left(9 \frac{d}{dx} \left(x\right)\right)}}{\sin{\left(9 x \right)}} + \frac{\frac{d}{dx} \left(\ln\left(x\right)\right)}{\ln\left(x\right)}$$

La dérivée du logarithme naturel est $$$\frac{d}{dx} \left(\ln\left(x\right)\right) = \frac{1}{x}$$$ :

$$\frac{9 \cos{\left(9 x \right)} \frac{d}{dx} \left(x\right)}{\sin{\left(9 x \right)}} + \frac{{\color{red}\left(\frac{d}{dx} \left(\ln\left(x\right)\right)\right)}}{\ln\left(x\right)} = \frac{9 \cos{\left(9 x \right)} \frac{d}{dx} \left(x\right)}{\sin{\left(9 x \right)}} + \frac{{\color{red}\left(\frac{1}{x}\right)}}{\ln\left(x\right)}$$

Appliquez la règle de puissance $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ avec $$$n = 1$$$, en d'autres termes, $$$\frac{d}{dx} \left(x\right) = 1$$$:

$$\frac{9 \cos{\left(9 x \right)} {\color{red}\left(\frac{d}{dx} \left(x\right)\right)}}{\sin{\left(9 x \right)}} + \frac{1}{x \ln\left(x\right)} = \frac{9 \cos{\left(9 x \right)} {\color{red}\left(1\right)}}{\sin{\left(9 x \right)}} + \frac{1}{x \ln\left(x\right)}$$

Simplifier:

$$\frac{9 \cos{\left(9 x \right)}}{\sin{\left(9 x \right)}} + \frac{1}{x \ln\left(x\right)} = \frac{9}{\tan{\left(9 x \right)}} + \frac{1}{x \ln\left(x\right)}$$

Ainsi, $$$\frac{d}{dx} \left(\ln\left(\ln\left(x\right)\right) + \ln\left(\sin{\left(9 x \right)}\right)\right) = \frac{9}{\tan{\left(9 x \right)}} + \frac{1}{x \ln\left(x\right)}$$$.

Ainsi, $$$\frac{\frac{d}{dx} \left(H{\left(x \right)}\right)}{H{\left(x \right)}} = \frac{9}{\tan{\left(9 x \right)}} + \frac{1}{x \ln\left(x\right)}$$$.

Donc, $$$\frac{d}{dx} \left(H{\left(x \right)}\right) = \left(\frac{9}{\tan{\left(9 x \right)}} + \frac{1}{x \ln\left(x\right)}\right) H{\left(x \right)} = 9 \ln\left(x\right) \cos{\left(9 x \right)} + \frac{\sin{\left(9 x \right)}}{x}$$$.