Question

if y=sqrt (sinx +y),then dy/dx is equal to

Answer 1

dy/dx=cosx/2y-1 the answer of the question is on the pic

Answer 2

Answer:  $\dfrac{dy}{dx}=\dfrac{\cos x}{2\sqrt{\sin x+y}-1}.$

Step-by-step explanation:  We are given the following equation :

$y=\sqrt{\sin x+y}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)$

We are to find the value of $\dfrac{dy}{dx}.$

We have from equation (i) that

$y=\sqrt{\sin x+y}\\\\\Rightarrow y^2=\sin x+y\\\\\Rightarrow y^2-y=\sin x.$

Differentiating both sides of the above equation with respect to x, we get

$\dfrac{d}{dx}(y^2-y)=\dfrac{d}{dx}\sin x\\\\\\\Rightarrow 2y\dfrac{dy}{dx}-\dfrac{dy}{dx}=\cos x\\\\\\\Rightarrow (2y-1)\dfrac{dy}{dx}=\cos x\\\\\\\Rightarrow \dfrac{dy}{dx}=\dfrac{\cos x}{2y-1}\\\\\\\Rightarrow \dfrac{dy}{dx}=\dfrac{\cos x}{2\sqrt{\sin x+y}-1}.$

Thus, we get

$\dfrac{dy}{dx}=\dfrac{\cos x}{2\sqrt{\sin x+y}-1}.$