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AI-তৈরি নমুনা উত্তর

ধরি,
$y = (\sqrt{x})^{\sqrt{x}} = (x^{1/2})^{\sqrt{x}} = x^{\frac{\sqrt{x}}{2}}$

উভয়পক্ষে $\ln$ নিয়ে পাই,
$\ln y = \ln\left(x^{\frac{\sqrt{x}}{2}}\right) = \frac{\sqrt{x}}{2} \ln x$

এখন, $x$ এর সাপেক্ষে অন্তরীকরণ করে পাই,
$\frac{1}{y} \frac{dy}{dx} = \frac{1}{2} \left[ \frac{d}{dx}(\sqrt{x}) \cdot \ln x + \sqrt{x} \cdot \frac{d}{dx}(\ln x) \right]$
$\Rightarrow \frac{1}{y} \frac{dy}{dx} = \frac{1}{2} \left[ \frac{1}{2\sqrt{x}} \ln x + \sqrt{x} \cdot \frac{1}{x} \right]$
$\Rightarrow \frac{1}{y} \frac{dy}{dx} = \frac{1}{2} \left[ \frac{\ln x}{2\sqrt{x}} + \frac{1}{\sqrt{x}} \right] = \frac{2 + \ln x}{4\sqrt{x}}$

$\therefore \frac{dy}{dx} = y \cdot \frac{2 + \ln x}{4\sqrt{x}} = (\sqrt{x})^{\sqrt{x}} \left( \frac{2 + \ln x}{4\sqrt{x}} \right)$
বা, $\frac{dy}{dx} = \frac{1}{4} x^{\frac{\sqrt{x}-1}{2}} (2 + \ln x)$