যদি x^2=5y^2+siny হয়,তাহলে dy/dx কত হবে? 

দেওয়া আছে, \(x^2 = 5y^2 + \sin y\)

আমরা \(x\) এর সাপেক্ষে উভয় দিকে অন্তরকলন করি।

\(\frac{d}{dx}(x^2) = \frac{d}{dx}(5y^2 + \sin y)\)

\(\implies 2x = \frac{d}{dx}(5y^2) + \frac{d}{dx}(\sin y)\)

\(\implies 2x = 5 \cdot \frac{d}{dx}(y^2) + \frac{d}{dx}(\sin y)\)

এখানে, \(\frac{d}{dx}(y^2) = \frac{d}{dy}(y^2) \cdot \frac{dy}{dx} = 2y \cdot \frac{dy}{dx}\) (chain rule)

এবং \(\frac{d}{dx}(\sin y) = \frac{d}{dy}(\sin y) \cdot \frac{dy}{dx} = \cos y \cdot \frac{dy}{dx}\) (chain rule)

সুতরাং, \(2x = 5 \cdot 2y \frac{dy}{dx} + \cos y \frac{dy}{dx}\)

\(\implies 2x = 10y \frac{dy}{dx} + \cos y \frac{dy}{dx}\)

\(\implies 2x = (10y + \cos y) \frac{dy}{dx}\)

\(\implies \frac{dy}{dx} = \frac{2x}{10y + \cos y}\)

অতএব, \(\frac{dy}{dx} = \frac{2x}{\cos y + 10y}\) 🥳