Best solutions to - "Second derivatives Find d2ydx2. 2x2+y2=4" - Plainmath

melodykap

Answered question

2021-10-13

Second derivatives Find $\frac{d^{2} y}{{d x}^{2}}$ . $2 x^{2} + y^{2} = 4$

Answer & Explanation

Ayesha Gomez

Skilled2021-10-14Added 104 answers

Step 1
To find

\frac{d^{2} y}{{d x}^{2}}

of

2 x^{2} + y^{2} = 4

Step 2

2 x^{2} + y^{2} = 4

On differentiating with respect to x we get:

4 x + 2 y \frac{d y}{d x} = 0

\frac{d y}{d x} = - \frac{4 x}{2 y}

\frac{d y}{d x} = - 2 \frac{x}{y}

Again differentiating with respect to x we get:

\frac{d^{2} y}{{d x}^{2}} = - 2 (\frac{y - x \frac{d y}{d x}}{y^{2}})

Now

\frac{d y}{d x} = - 2 \frac{x}{y}

Therefore

\frac{d^{2} y}{{d x}^{2}} = - 2 (\frac{y + 2 x \cdot \frac{x}{y}}{y^{2}})

\frac{d^{2} y}{{d x}^{2}} = - 2 (\frac{y + 2 x \cdot \frac{x}{y}}{y^{2}}) = - 2 (\frac{y^{2} + 2 x^{2}}{y^{3}})

As

2 x^{2} + y^{2} = 4

Therefore,

\frac{d^{2} y}{{d x}^{2}} = - \frac{8}{y^{3}}

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