Derivative of $$$\ln\left(\sqrt{10} \sqrt{x}\right)$$$
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Find $$$\frac{d}{dx} \left(\ln\left(\sqrt{10} \sqrt{x}\right)\right)$$$.
Solution
The function $$$\ln\left(\sqrt{10} \sqrt{x}\right)$$$ is the composition $$$f{\left(g{\left(x \right)} \right)}$$$ of two functions $$$f{\left(u \right)} = \ln\left(u\right)$$$ and $$$g{\left(x \right)} = \sqrt{10} \sqrt{x}$$$.
Apply the chain rule $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:
$${\color{red}\left(\frac{d}{dx} \left(\ln\left(\sqrt{10} \sqrt{x}\right)\right)\right)} = {\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right) \frac{d}{dx} \left(\sqrt{10} \sqrt{x}\right)\right)}$$The derivative of the natural logarithm is $$$\frac{d}{du} \left(\ln\left(u\right)\right) = \frac{1}{u}$$$:
$${\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right)\right)} \frac{d}{dx} \left(\sqrt{10} \sqrt{x}\right) = {\color{red}\left(\frac{1}{u}\right)} \frac{d}{dx} \left(\sqrt{10} \sqrt{x}\right)$$Return to the old variable:
$$\frac{\frac{d}{dx} \left(\sqrt{10} \sqrt{x}\right)}{{\color{red}\left(u\right)}} = \frac{\frac{d}{dx} \left(\sqrt{10} \sqrt{x}\right)}{{\color{red}\left(\sqrt{10} \sqrt{x}\right)}}$$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 $$$c = \sqrt{10}$$$ and $$$f{\left(x \right)} = \sqrt{x}$$$:
$$\frac{\sqrt{10} {\color{red}\left(\frac{d}{dx} \left(\sqrt{10} \sqrt{x}\right)\right)}}{10 \sqrt{x}} = \frac{\sqrt{10} {\color{red}\left(\sqrt{10} \frac{d}{dx} \left(\sqrt{x}\right)\right)}}{10 \sqrt{x}}$$Apply the power rule $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ with $$$n = \frac{1}{2}$$$:
$$\frac{{\color{red}\left(\frac{d}{dx} \left(\sqrt{x}\right)\right)}}{\sqrt{x}} = \frac{{\color{red}\left(\frac{1}{2 \sqrt{x}}\right)}}{\sqrt{x}}$$Thus, $$$\frac{d}{dx} \left(\ln\left(\sqrt{10} \sqrt{x}\right)\right) = \frac{1}{2 x}$$$.
Answer
$$$\frac{d}{dx} \left(\ln\left(\sqrt{10} \sqrt{x}\right)\right) = \frac{1}{2 x}$$$A