If y = sin x + y , then d y d x is equal to:

Question

If $y = \sqrt{\sin x + y}$ , then $\frac{d y}{d x}$ is equal to:

Smart Achievers · Accepted Answer

(1)

Ans. $\frac{c o s x}{2 y - 1}$

Explanation:

$∵$ $y = {(\sin x + y)}^{^{\frac{1}{2}}}$

$∴$ $\frac{d y}{d x} = \frac{1}{2} {(\sin x + y)}^{- \frac{1}{2}} \cdot \frac{d}{d x} (\sin x + y)$ [By chain rule derivative]

$\Rightarrow$ $\frac{d y}{d x} = \frac{1}{2} \cdot \frac{1}{{(\sin x + y)}^{1 / 2}} (\cos x + \frac{d y}{d x})$

$\Rightarrow$ $\frac{d y}{d x} = \frac{1}{2 y} (\cos x + \frac{d y}{d x})$ $[∵ {(\sin x + y)}^{1 / 2} = y]$

$\Rightarrow$ $\frac{d y}{d x} (1 - \frac{1}{2 y}) = \frac{\cos x}{2 y}$

$∴$ $\frac{d y}{d x} = \frac{\cos x}{2 y} \cdot \frac{2 y}{2 y - 1} = \frac{\cos x}{2 y - 1}$

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