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

Correct Answer is : cosx2y−1cosx/2y-12y−1cosx

Correct Answer is : cosx2y−1cosx/2y-12y−1cosx

Test Index

AP DSC School Assistant Math Model Test 1

Section: Subject Specific

Question : 72 of 160

Marks: +1, -0

If y =

\sqrt{sinx+y}

\frac{dy}{dx}

is equal to

Solution:

Concept:

The derivative of a square root function requires the chain rule:

\frac{d}{dx}\sqrt{u} = \frac{1}{2\sqrt{u}} \cdot \frac{du}{dx}

. Here

u = \sin x + y

, and

y

itself is a function of

x

, so we differentiate implicitly.

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 \left(\cos x + \frac{dy}{dx}\right)

Since

y = \sqrt{\sin x + y}

, we replace

\sqrt{\sin x + y}

with

y

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

Multiply both sides by

2y

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

Bring

\frac{dy}{dx}

terms together:

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

Factor

\frac{dy}{dx}

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

Hence,

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

Answer:

Option A:

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

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