যদি y=(2/(root(n)(x)))^(1/x) হয়, তবে dy/dx এর মান কোনটি? 

প্রশ্ন: যদি \(y=\left(\frac{2}{\sqrt[n]{x}}\right)^{\frac{1}{x}}\) হয়, তবে \(\frac{dy}{dx}\) এর মান কোনটি?

উত্তর: \( \left(\frac{2}{\sqrt[n]{x}}\right)^{\frac{1}{x}} \left[ -\frac{1}{x^2} \ln\left(\frac{2}{\sqrt[n]{x}}\right) - \frac{1}{nx^2} \right] \)

সমাধান:

ধরি, \(y=\left(\frac{2}{\sqrt[n]{x}}\right)^{\frac{1}{x}}\)

উভয় পক্ষে লন নিয়ে পাই,

\(\ln y = \ln \left(\frac{2}{\sqrt[n]{x}}\right)^{\frac{1}{x}}\)

\(\ln y = \frac{1}{x} \ln \left(\frac{2}{\sqrt[n]{x}}\right)\)

\(\ln y = \frac{1}{x} \left[ \ln 2 - \ln \sqrt[n]{x} \right]\)

\(\ln y = \frac{1}{x} \left[ \ln 2 - \ln x^{\frac{1}{n}} \right]\)

\(\ln y = \frac{1}{x} \left[ \ln 2 - \frac{1}{n} \ln x \right]\)

\(\ln y = \frac{\ln 2}{x} - \frac{\ln x}{nx}\)

এখন, x এর সাপেক্ষে উভয় পক্ষে অন্তরকলন করে পাই,

\(\frac{1}{y} \frac{dy}{dx} = \frac{d}{dx} \left( \frac{\ln 2}{x} - \frac{\ln x}{nx} \right)\)

\(\frac{1}{y} \frac{dy}{dx} = \ln 2 \frac{d}{dx} \left( \frac{1}{x} \right) - \frac{1}{n} \frac{d}{dx} \left( \frac{\ln x}{x} \right)\)

\(\frac{1}{y} \frac{dy}{dx} = \ln 2 \left( -\frac{1}{x^2} \right) - \frac{1}{n} \left[ \frac{x \cdot \frac{1}{x} - \ln x \cdot 1}{x^2} \right]\)

\(\frac{1}{y} \frac{dy}{dx} = -\frac{\ln 2}{x^2} - \frac{1}{n} \left[ \frac{1 - \ln x}{x^2} \right]\)

\(\frac{1}{y} \frac{dy}{dx} = -\frac{\ln 2}{x^2} - \frac{1}{nx^2} + \frac{\ln x}{nx^2}\)

\(\frac{1}{y} \frac{dy}{dx} = \frac{1}{x^2} \left[ -\ln 2 - \frac{1}{n} + \frac{\ln x}{n} \right]\)

\(\frac{dy}{dx} = y \cdot \frac{1}{x^2} \left[ -\ln 2 - \frac{1}{n} + \frac{\ln x}{n} \right]\)

\(\frac{dy}{dx} = \left(\frac{2}{\sqrt[n]{x}}\right)^{\frac{1}{x}} \cdot \frac{1}{x^2} \left[ \frac{\ln x}{n} - \ln 2 - \frac{1}{n} \right]\)

\(\frac{dy}{dx} = \left(\frac{2}{\sqrt[n]{x}}\right)^{\frac{1}{x}} \cdot \left[ -\frac{\ln 2}{x^2} + \frac{\ln x -1}{nx^2} \right]\)

\(\frac{dy}{dx} = \left(\frac{2}{\sqrt[n]{x}}\right)^{\frac{1}{x}} \cdot \left[ -\frac{1}{x^2} \ln 2 + \frac{1}{nx^2} (\ln x -1) \right]\)

\(\frac{dy}{dx} = \left(\frac{2}{\sqrt[n]{x}}\right)^{\frac{1}{x}} \left[ -\frac{1}{x^2} \ln\left(\frac{2}{\sqrt[n]{x}}\right) - \frac{1}{nx^2} \right] \)