Question

Ayuda con este problemas de derivadas implícitas

Answer 1

$\dfrac{d}{dx}\cos(x+y)=\dfrac{d}{dx}[y^2\sin x]\\\\\\\dfrac{d(x+y)}{dx}\cdot\dfrac{d\cos(x+y)}{d(x+y)}=\dfrac{dy^2}{dx}\cdot \sin x+y^2\cdot\dfrac{d\sin x}{dx}\\ \\ \\\\(\dfrac{dx}{dx}+\dfrac{dy}{dx})[-\sin(x+y)]=(\dfrac{dy^2}{dy}\cdot \dfrac{dy}{dx})\sin x+y^2\cos x\\\\\\-(1+\dfrac{dy}{dx})\sin (x+y)=2y\cdot\dfrac{dy}{dx}\cdot\sin x+y^2\cos x\\\\\\\text{por practicidad pongamos }y'=\dfrac{dy}{dx}\\\\\\-(1+y')\sin (x+y)=2y\cdot y'\cdot\sin x+y^2\cos x$

$-\sin (x+y)-y'\sin (x+y)=(2y \cdot\sin x)y'+y^2\cos x\\\\-\sin (x+y)-y^2\cos x=[2y \cdot\sin x+\sin (x+y)]y'\\\\y'=-\dfrac{\sin (x+y)+y^2\cos x}{2y \sin x+\sin (x+y)}$