Derivative of $$$\ln\left(2 x^{3}\right)$$$
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Find $$$\frac{d}{dx} \left(\ln\left(2 x^{3}\right)\right)$$$.
Solution
The function $$$\ln\left(2 x^{3}\right)$$$ is the composition $$$f{\left(g{\left(x \right)} \right)}$$$ of two functions $$$f{\left(u \right)} = \ln\left(u\right)$$$ and $$$g{\left(x \right)} = 2 x^{3}$$$.
Apply the chain rule $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:
$${\color{red}\left(\frac{d}{dx} \left(\ln\left(2 x^{3}\right)\right)\right)} = {\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right) \frac{d}{dx} \left(2 x^{3}\right)\right)}$$The derivative of the natural logarithm is $$$\frac{d}{du} \left(\ln\left(u\right)\right) = \frac{1}{u}$$$:
$${\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right)\right)} \frac{d}{dx} \left(2 x^{3}\right) = {\color{red}\left(\frac{1}{u}\right)} \frac{d}{dx} \left(2 x^{3}\right)$$Return to the old variable:
$$\frac{\frac{d}{dx} \left(2 x^{3}\right)}{{\color{red}\left(u\right)}} = \frac{\frac{d}{dx} \left(2 x^{3}\right)}{{\color{red}\left(2 x^{3}\right)}}$$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 $$$f{\left(x \right)} = x^{3}$$$:
$$\frac{{\color{red}\left(\frac{d}{dx} \left(2 x^{3}\right)\right)}}{2 x^{3}} = \frac{{\color{red}\left(2 \frac{d}{dx} \left(x^{3}\right)\right)}}{2 x^{3}}$$Apply the power rule $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ with $$$n = 3$$$:
$$\frac{{\color{red}\left(\frac{d}{dx} \left(x^{3}\right)\right)}}{x^{3}} = \frac{{\color{red}\left(3 x^{2}\right)}}{x^{3}}$$Thus, $$$\frac{d}{dx} \left(\ln\left(2 x^{3}\right)\right) = \frac{3}{x}$$$.
Answer
$$$\frac{d}{dx} \left(\ln\left(2 x^{3}\right)\right) = \frac{3}{x}$$$A