y=(1/root(n)(x))^(1/x) হলে dy/dx এর মান কোনটি?

Explanation:

Another Explanation (5): দেওয়া আছে, \(y = \left(\frac{1}{\sqrt[n]{x}}\right)^{\frac{1}{x}}\) এখন, \(y = \left(x^{-\frac{1}{n}}\right)^{\frac{1}{x}} = x^{-\frac{1}{nx}}\) উভয় পক্ষে লন (ln) নিয়ে পাই, \(\ln y = \ln \left(x^{-\frac{1}{nx}}\right)\) \(\ln y = -\frac{1}{nx} \ln x\) এখন, x এর সাপেক্ষে অন্তরীকরণ করে পাই, \(\frac{1}{y} \frac{dy}{dx} = -\frac{1}{n} \frac{d}{dx} \left(\frac{\ln x}{x}\right)\) \(\frac{1}{y} \frac{dy}{dx} = -\frac{1}{n} \left[\frac{x \cdot \frac{1}{x} - \ln x \cdot 1}{x^2}\right]\) 😃 \(\frac{1}{y} \frac{dy}{dx} = -\frac{1}{n} \left[\frac{1 - \ln x}{x^2}\right]\) \(\frac{1}{y} \frac{dy}{dx} = \frac{1}{n} \left[\frac{\ln x - 1}{x^2}\right]\) অতএব, \(\frac{dy}{dx} = y \cdot \frac{1}{n} \left[\frac{\ln x - 1}{x^2}\right]\) \(\frac{dy}{dx} = \frac{y(\ln x - 1)}{nx^2}\) 🎉 সুতরাং, \(\frac{dy}{dx} = \frac{y(\ln x - 1)}{nx^2}\) 💖