If y = { x+y}, then dy/dx is equal to

Correct Answer is : cos⁡x2y−1 x/2 y−12y−1cosx

Correct Answer is : cos⁡x2y−1 x/2 y−12y−1cosx

Test Index

AP DSC School Assistant Math Model Test 3

Section: Subject Specific

Question : 112 of 160

Marks: +1, -0

If y =

\rm \sqrt{\sin x+y}

\rm \frac{dy}{dx}

is equal to

Solution:

Concept:

This problem involves implicit differentiation because the variable

y

appears on both sides of the equation. The derivative of

\sqrt{u}

\frac{1}{2\sqrt{u}}\frac{du}{dx}

Explanation:

Given

y = \sqrt{\sin x + y}

. Differentiate both sides with respect to

x

\frac{dy}{dx} = \frac{1}{2\sqrt{\sin x + y}} \cdot (\cos x + \frac{dy}{dx})

Since

\sqrt{\sin x + y} = y

, substitute

y

for the square root:

\frac{dy}{dx} = \frac{1}{2y} (\cos x + \frac{dy}{dx})

Multiply both sides by

2y

2y\frac{dy}{dx} = \cos x + \frac{dy}{dx}

Bring

\frac{dy}{dx}

terms to one side:

2y\frac{dy}{dx} - \frac{dy}{dx} = \cos x

⇒

(2y-1)\frac{dy}{dx} = \cos x

Therefore,

\frac{dy}{dx} = \frac{\cos x}{2y-1}

Answer:

Option A:

\frac{\cos x}{2y-1}

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