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Derivada de $$$x^{2} \sin{\left(x \right)}$$$
Calculadora relacionada: Calculadora de Derivativos
Sua entrada
Encontre $$$\frac{d}{dx} \left(x^{2} \sin{\left(x \right)}\right)$$$.
Solução
Seja $$$H{\left(x \right)} = x^{2} \sin{\left(x \right)}$$$.
Pegue o logaritmo de ambos os lados: $$$\ln\left(H{\left(x \right)}\right) = \ln\left(x^{2} \sin{\left(x \right)}\right)$$$.
Reescreva o RHS usando as propriedades dos logaritmos: $$$\ln\left(H{\left(x \right)}\right) = 2 \ln\left(x\right) + \ln\left(\sin{\left(x \right)}\right)$$$.
Diferencie separadamente os dois lados da equação: $$$\frac{d}{dx} \left(\ln\left(H{\left(x \right)}\right)\right) = \frac{d}{dx} \left(2 \ln\left(x\right) + \ln\left(\sin{\left(x \right)}\right)\right)$$$.
Diferencie o LHS da equação.
A função $$$\ln\left(H{\left(x \right)}\right)$$$ é a composição $$$f{\left(g{\left(x \right)} \right)}$$$ de duas funções $$$f{\left(u \right)} = \ln\left(u\right)$$$ e $$$g{\left(x \right)} = H{\left(x \right)}$$$.
Aplique a regra da cadeia $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:
$${\color{red}\left(\frac{d}{dx} \left(\ln\left(H{\left(x \right)}\right)\right)\right)} = {\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right) \frac{d}{dx} \left(H{\left(x \right)}\right)\right)}$$A derivada do logaritmo natural é $$$\frac{d}{du} \left(\ln\left(u\right)\right) = \frac{1}{u}$$$:
$${\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right)\right)} \frac{d}{dx} \left(H{\left(x \right)}\right) = {\color{red}\left(\frac{1}{u}\right)} \frac{d}{dx} \left(H{\left(x \right)}\right)$$Volte para a variável antiga:
$$\frac{\frac{d}{dx} \left(H{\left(x \right)}\right)}{{\color{red}\left(u\right)}} = \frac{\frac{d}{dx} \left(H{\left(x \right)}\right)}{{\color{red}\left(H{\left(x \right)}\right)}}$$Assim, $$$\frac{d}{dx} \left(\ln\left(H{\left(x \right)}\right)\right) = \frac{\frac{d}{dx} \left(H{\left(x \right)}\right)}{H{\left(x \right)}}$$$.
Diferencie o RHS da equação.
A derivada de uma soma/diferença é a soma/diferença das derivadas:
$${\color{red}\left(\frac{d}{dx} \left(2 \ln\left(x\right) + \ln\left(\sin{\left(x \right)}\right)\right)\right)} = {\color{red}\left(\frac{d}{dx} \left(2 \ln\left(x\right)\right) + \frac{d}{dx} \left(\ln\left(\sin{\left(x \right)}\right)\right)\right)}$$Aplique a regra múltipla constante $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ com $$$c = 2$$$ e $$$f{\left(x \right)} = \ln\left(x\right)$$$:
$${\color{red}\left(\frac{d}{dx} \left(2 \ln\left(x\right)\right)\right)} + \frac{d}{dx} \left(\ln\left(\sin{\left(x \right)}\right)\right) = {\color{red}\left(2 \frac{d}{dx} \left(\ln\left(x\right)\right)\right)} + \frac{d}{dx} \left(\ln\left(\sin{\left(x \right)}\right)\right)$$A função $$$\ln\left(\sin{\left(x \right)}\right)$$$ é a composição $$$f{\left(g{\left(x \right)} \right)}$$$ de duas funções $$$f{\left(u \right)} = \ln\left(u\right)$$$ e $$$g{\left(x \right)} = \sin{\left(x \right)}$$$.
Aplique a regra da cadeia $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:
$${\color{red}\left(\frac{d}{dx} \left(\ln\left(\sin{\left(x \right)}\right)\right)\right)} + 2 \frac{d}{dx} \left(\ln\left(x\right)\right) = {\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right) \frac{d}{dx} \left(\sin{\left(x \right)}\right)\right)} + 2 \frac{d}{dx} \left(\ln\left(x\right)\right)$$A derivada do logaritmo natural é $$$\frac{d}{du} \left(\ln\left(u\right)\right) = \frac{1}{u}$$$:
$${\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right)\right)} \frac{d}{dx} \left(\sin{\left(x \right)}\right) + 2 \frac{d}{dx} \left(\ln\left(x\right)\right) = {\color{red}\left(\frac{1}{u}\right)} \frac{d}{dx} \left(\sin{\left(x \right)}\right) + 2 \frac{d}{dx} \left(\ln\left(x\right)\right)$$Volte para a variável antiga:
$$2 \frac{d}{dx} \left(\ln\left(x\right)\right) + \frac{\frac{d}{dx} \left(\sin{\left(x \right)}\right)}{{\color{red}\left(u\right)}} = 2 \frac{d}{dx} \left(\ln\left(x\right)\right) + \frac{\frac{d}{dx} \left(\sin{\left(x \right)}\right)}{{\color{red}\left(\sin{\left(x \right)}\right)}}$$A derivada do seno é $$$\frac{d}{dx} \left(\sin{\left(x \right)}\right) = \cos{\left(x \right)}$$$:
$$2 \frac{d}{dx} \left(\ln\left(x\right)\right) + \frac{{\color{red}\left(\frac{d}{dx} \left(\sin{\left(x \right)}\right)\right)}}{\sin{\left(x \right)}} = 2 \frac{d}{dx} \left(\ln\left(x\right)\right) + \frac{{\color{red}\left(\cos{\left(x \right)}\right)}}{\sin{\left(x \right)}}$$A derivada do logaritmo natural é $$$\frac{d}{dx} \left(\ln\left(x\right)\right) = \frac{1}{x}$$$:
$$2 {\color{red}\left(\frac{d}{dx} \left(\ln\left(x\right)\right)\right)} + \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} = 2 {\color{red}\left(\frac{1}{x}\right)} + \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}$$Simplificar:
$$\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{2}{x} = \frac{\frac{x}{\tan{\left(x \right)}} + 2}{x}$$Assim, $$$\frac{d}{dx} \left(2 \ln\left(x\right) + \ln\left(\sin{\left(x \right)}\right)\right) = \frac{\frac{x}{\tan{\left(x \right)}} + 2}{x}$$$.
Portanto, $$$\frac{\frac{d}{dx} \left(H{\left(x \right)}\right)}{H{\left(x \right)}} = \frac{\frac{x}{\tan{\left(x \right)}} + 2}{x}$$$.
Portanto, $$$\frac{d}{dx} \left(H{\left(x \right)}\right) = \frac{\left(\frac{x}{\tan{\left(x \right)}} + 2\right) H{\left(x \right)}}{x} = x \left(x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right)$$$.
Responder
$$$\frac{d}{dx} \left(x^{2} \sin{\left(x \right)}\right) = x \left(x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right)$$$A