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Function 2x+3y=sinx, then the value of d...

Function 2x+3y=sinx, then the value of dy/dx is:

A

`-((cosx+2))/2`

B

`(cosx-2)/3`

C

cosx-2

D

`(cosx-3)/2`

Text Solution

AI Generated Solution

The correct Answer is:
To find the value of \(\frac{dy}{dx}\) for the equation \(2x + 3y = \sin x\), we will follow these steps: ### Step 1: Rearrange the equation Start with the given equation: \[ 2x + 3y = \sin x \] We can rearrange this to isolate \(3y\): \[ 3y = \sin x - 2x \] ### Step 2: Differentiate both sides with respect to \(x\) Now, we differentiate both sides of the equation with respect to \(x\): \[ \frac{d}{dx}(3y) = \frac{d}{dx}(\sin x - 2x) \] Using the chain rule on the left side and the standard differentiation rules on the right side, we get: \[ 3 \frac{dy}{dx} = \cos x - 2 \] ### Step 3: Solve for \(\frac{dy}{dx}\) Now, we need to isolate \(\frac{dy}{dx}\): \[ \frac{dy}{dx} = \frac{\cos x - 2}{3} \] ### Final Answer Thus, the value of \(\frac{dy}{dx}\) is: \[ \frac{dy}{dx} = \frac{\cos x - 2}{3} \] ---
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