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Derivative of $$$x^{2} \sin{\left(x \right)}$$$

The calculator will find the derivative of $$$x^{2} \sin{\left(x \right)}$$$ using the logarithmic differentiation, with steps shown.

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Find $$$\frac{d}{dx} \left(x^{2} \sin{\left(x \right)}\right)$$$.

Solution

Let $$$H{\left(x \right)} = x^{2} \sin{\left(x \right)}$$$.

Take the logarithm of both sides: $$$\ln\left(H{\left(x \right)}\right) = \ln\left(x^{2} \sin{\left(x \right)}\right)$$$.

Rewrite the RHS using the properties of logarithms: $$$\ln\left(H{\left(x \right)}\right) = 2 \ln\left(x\right) + \ln\left(\sin{\left(x \right)}\right)$$$.

Differentiate separately both sides of the equation: $$$\frac{d}{dx} \left(\ln\left(H{\left(x \right)}\right)\right) = \frac{d}{dx} \left(2 \ln\left(x\right) + \ln\left(\sin{\left(x \right)}\right)\right)$$$.

Differentiate the LHS of the equation.

The function $$$\ln\left(H{\left(x \right)}\right)$$$ is the composition $$$f{\left(g{\left(x \right)} \right)}$$$ of two functions $$$f{\left(u \right)} = \ln\left(u\right)$$$ and $$$g{\left(x \right)} = H{\left(x \right)}$$$.

Apply the chain rule $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:

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

The derivative of the natural logarithm is $$$\frac{d}{du} \left(\ln\left(u\right)\right) = \frac{1}{u}$$$:

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

Return to the old variable:

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

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

Differentiate the RHS of the equation.

The derivative of a sum/difference is the sum/difference of derivatives:

$${\color{red}\left(\frac{d}{dx} \left(2 \ln\left(x\right) + \ln\left(\sin{\left(x \right)}\right)\right)\right)} = {\color{red}\left(\frac{d}{dx} \left(2 \ln\left(x\right)\right) + \frac{d}{dx} \left(\ln\left(\sin{\left(x \right)}\right)\right)\right)}$$

Apply the constant multiple rule $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ with $$$c = 2$$$ and $$$f{\left(x \right)} = \ln\left(x\right)$$$:

$${\color{red}\left(\frac{d}{dx} \left(2 \ln\left(x\right)\right)\right)} + \frac{d}{dx} \left(\ln\left(\sin{\left(x \right)}\right)\right) = {\color{red}\left(2 \frac{d}{dx} \left(\ln\left(x\right)\right)\right)} + \frac{d}{dx} \left(\ln\left(\sin{\left(x \right)}\right)\right)$$

The derivative of the natural logarithm is $$$\frac{d}{dx} \left(\ln\left(x\right)\right) = \frac{1}{x}$$$:

$$2 {\color{red}\left(\frac{d}{dx} \left(\ln\left(x\right)\right)\right)} + \frac{d}{dx} \left(\ln\left(\sin{\left(x \right)}\right)\right) = 2 {\color{red}\left(\frac{1}{x}\right)} + \frac{d}{dx} \left(\ln\left(\sin{\left(x \right)}\right)\right)$$

The function $$$\ln\left(\sin{\left(x \right)}\right)$$$ is the composition $$$f{\left(g{\left(x \right)} \right)}$$$ of two functions $$$f{\left(u \right)} = \ln\left(u\right)$$$ and $$$g{\left(x \right)} = \sin{\left(x \right)}$$$.

Apply the chain rule $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:

$${\color{red}\left(\frac{d}{dx} \left(\ln\left(\sin{\left(x \right)}\right)\right)\right)} + \frac{2}{x} = {\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right) \frac{d}{dx} \left(\sin{\left(x \right)}\right)\right)} + \frac{2}{x}$$

The derivative of the natural logarithm is $$$\frac{d}{du} \left(\ln\left(u\right)\right) = \frac{1}{u}$$$:

$${\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right)\right)} \frac{d}{dx} \left(\sin{\left(x \right)}\right) + \frac{2}{x} = {\color{red}\left(\frac{1}{u}\right)} \frac{d}{dx} \left(\sin{\left(x \right)}\right) + \frac{2}{x}$$

Return to the old variable:

$$\frac{\frac{d}{dx} \left(\sin{\left(x \right)}\right)}{{\color{red}\left(u\right)}} + \frac{2}{x} = \frac{\frac{d}{dx} \left(\sin{\left(x \right)}\right)}{{\color{red}\left(\sin{\left(x \right)}\right)}} + \frac{2}{x}$$

The derivative of the sine is $$$\frac{d}{dx} \left(\sin{\left(x \right)}\right) = \cos{\left(x \right)}$$$:

$$\frac{{\color{red}\left(\frac{d}{dx} \left(\sin{\left(x \right)}\right)\right)}}{\sin{\left(x \right)}} + \frac{2}{x} = \frac{{\color{red}\left(\cos{\left(x \right)}\right)}}{\sin{\left(x \right)}} + \frac{2}{x}$$

Simplify:

$$\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{2}{x} = \frac{\frac{x}{\tan{\left(x \right)}} + 2}{x}$$

Thus, $$$\frac{d}{dx} \left(2 \ln\left(x\right) + \ln\left(\sin{\left(x \right)}\right)\right) = \frac{\frac{x}{\tan{\left(x \right)}} + 2}{x}$$$.

Hence, $$$\frac{\frac{d}{dx} \left(H{\left(x \right)}\right)}{H{\left(x \right)}} = \frac{\frac{x}{\tan{\left(x \right)}} + 2}{x}$$$.

Therefore, $$$\frac{d}{dx} \left(H{\left(x \right)}\right) = \frac{\left(\frac{x}{\tan{\left(x \right)}} + 2\right) H{\left(x \right)}}{x} = x \left(x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right)$$$.

Answer

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

Related Note: Logarithmic Differentiation