Question

Find dy/dx
√(x+y) = x - 2y

Answer 1

`\bf √(x+y)=x-2y\implies (x+y)^{(1)/(2)}=x-2y \\\\\\ \cfrac{1}{2}(x+y)^{-(1)/(2)}\left(1+(dy)/(dx) \right)=1-2(dy)/(dx) \implies \cfrac{1+(dy)/(dx) }{2(x+y)^{(1)/(2)}}=1-2(dy)/(dx) \\\\\\ 1+(dy)/(dx) =2(x+y)^{(1)/(2)}-2(x+y)^{(1)/(2)}\cdot 2(dy)/(dx) \\\\\\ 1+(dy)/(dx) =2(x+y)^{(1)/(2)}-4(x+y)^{(1)/(2)}(dy)/(dx) \\\\\\ (dy)/(dx)+4(x+y)^{(1)/(2)}(dy)/(dx) =2(x+y)^{(1)/(2)}-1`


`\bf \cfrac{dy}{dx}\left[ 1+4(x+y)^{(1)/(2)} \right]=2(x+y)^{(1)/(2)}-1\implies \cfrac{dy}{dx}=\cfrac{2(x+y)^{(1)/(2)}-1}{1+4(x+y)^{(1)/(2)} } \\\\\\ \cfrac{dy}{dx}=\cfrac{2√(x+y)-1}{1+4√(x+y)}`