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Given that \(y^2 + xy = 5,find \frac{dy}{dx}\)...

Given that \(y^2 + xy = 5,find \frac{dy}{dx}\)

A.

\(\frac{y}{2y + x}\)

B.

\(\frac{-y}{2y + x}\)

C.

\(\frac{-y}{2y - x}\)

D.

\(\frac{y}{2y + x}\)

Correct answer is B

\(y^2 + xy = 5\)

By implicit differentiation

\(=2y\frac{dy}{dx}+y+x\frac{dy}{dx}=0\)

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

Factor out \(\frac{dy}{dx}\)

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

\(∴\frac{dy}{dx}=\frac{-y}{2y + x}\)