d/dx (cot(2√x))=কত?

Another Explanation (5): প্রশ্ন: \(\frac{d}{dx} \left( \cot(2\sqrt{x}) \right) = \text{কত?}\) উত্তর: \(\displaystyle \frac{-\csc^2(2\sqrt{x})}{\sqrt{x}}\) সমাধান: প্রথমে, ধরা যাক \( y = \cot(2\sqrt{x}) \) আমরা চাই \( \frac{dy}{dx} \) প্রথমে, চেইন রুল ব্যবহার করে: \[ \frac{dy}{dx} = \frac{d}{dx} \left( \cot(u) \right) \quad \text{যেখানে} \quad u = 2 \sqrt{x} \] এবং, \[ \frac{d}{dx} \left( \cot(u) \right) = -\csc^2(u) \cdot \frac{du}{dx} \] এবং, \[ u = 2 \sqrt{x} = 2 x^{1/2} \] তাহলে, \[ \frac{du}{dx} = 2 \times \frac{1}{2} x^{-1/2} = x^{-1/2} = \frac{1}{\sqrt{x}} \] অতএব, \[ \frac{dy}{dx} = -\csc^2(2 \sqrt{x}) \times \frac{1}{\sqrt{x}} \] অর্থাৎ, \[ \boxed{\frac{d}{dx} \left( \cot(2\sqrt{x}) \right) = -\frac{\csc^2(2\sqrt{x})}{\sqrt{x}}} \]