Derivative of $$$\sqrt{2} \sqrt{x} \ln\left(3\right)$$$
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Your Input
Find $$$\frac{d}{dx} \left(\sqrt{2} \sqrt{x} \ln\left(3\right)\right)$$$.
Solution
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 $$$c = \sqrt{2} \ln\left(3\right)$$$ and $$$f{\left(x \right)} = \sqrt{x}$$$:
$${\color{red}\left(\frac{d}{dx} \left(\sqrt{2} \sqrt{x} \ln\left(3\right)\right)\right)} = {\color{red}\left(\sqrt{2} \ln\left(3\right) \frac{d}{dx} \left(\sqrt{x}\right)\right)}$$Apply the power rule $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ with $$$n = \frac{1}{2}$$$:
$$\sqrt{2} \ln\left(3\right) {\color{red}\left(\frac{d}{dx} \left(\sqrt{x}\right)\right)} = \sqrt{2} \ln\left(3\right) {\color{red}\left(\frac{1}{2 \sqrt{x}}\right)}$$Thus, $$$\frac{d}{dx} \left(\sqrt{2} \sqrt{x} \ln\left(3\right)\right) = \frac{\sqrt{2} \ln\left(3\right)}{2 \sqrt{x}}$$$.
Answer
$$$\frac{d}{dx} \left(\sqrt{2} \sqrt{x} \ln\left(3\right)\right) = \frac{\sqrt{2} \ln\left(3\right)}{2 \sqrt{x}}$$$A