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Implicit Differentiation

If x2+y2=4,...

Question

If $x^{2} + y^{2} = 4$ , then $y \frac{d y}{d x} + x$ is equal to

A

4

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B

0

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C

1

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D

- 1

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Solution

The correct option is D $0$
Given $x^{2} + y^{2} = 4$
On differentiating w.r.t $x$ ,we get
$2 x + 2 y \frac{d y}{d x} = 0$
$\Rightarrow$ $y \frac{d y}{d x} + x = 0$

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y \frac{d y}{d x} = x ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \frac{y^{2}}{x^{2}} + \frac{ϕ (\frac{y^{2}}{x^{2}})}{ϕ^{'} (\frac{y^{2}}{x^{2}})} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦,

x > 0,

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y (1) = - 1,

ϕ (\frac{y^{2}}{4})

x {log}_{e} ({log}_{e} x) - x^{2} + y^{2} = 4 (y > 0)

\frac{d y}{d x}

x = e

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{cos}^{- 1} (\frac{x^{2} - y^{2}}{x^{2} + y^{2}}) = 2 k

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\frac{d y}{d x}

x = e

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