Lipossig

2021-03-31

Consider the equation ${y}^{3}-\frac{3}{2}{x}^{2}=1$ .

jlo2niT

Skilled2021-04-02Added 96 answers

Step 1

Consider the equation

${y}^{3}-\frac{3}{2}{x}^{2}=1$

Let us find the second derivative implicitly.

Step 2

Differentiate each term with respect to x

$\frac{d}{dx}({y}^{3}-\frac{3}{2}{x}^{2})=\frac{d}{dx}$ ...(1)

$\frac{d}{dx}\left({y}^{3}\right)-\frac{3}{2}\frac{d}{dx}\left({x}^{2}\right)=0$

$3{y}^{2}\frac{dy}{dx}-3x=0$

$\frac{dy}{dx}=\frac{x}{{y}^{2}}\to \left(1\right)$

Step 3

Differentiate second time with respect to x

$\frac{{d}^{2}y}{{dx}^{2}}=\frac{d}{dx}\left(\frac{x}{{y}^{2}}\right)$

$=\frac{\frac{d}{dx}\left(x\right){y}^{2}-\frac{d}{dx}\left({y}^{2}x\right)}{{\left({y}^{2}\right)}^{2}}$

$=\frac{1\cdot {y}^{2}-2y\frac{dy}{dx}x}{{y}^{4}}$

$=\frac{y-2x\frac{dy}{dx}}{{y}^{3}}$

Step 4

From equation (1) we have$\frac{dy}{dx}=\frac{x}{{y}^{2}}$ . Substituting this we get

$\frac{{d}^{2}y}{{dx}^{2}}=\frac{y-2x\left(\frac{x}{{y}^{2}}\right)}{{y}^{3}}$

$=\frac{{y}^{3}-2{x}^{2}}{{y}^{5}}$

Consider the equation

Let us find the second derivative implicitly.

Step 2

Differentiate each term with respect to x

Step 3

Differentiate second time with respect to x

Step 4

From equation (1) we have

Jeffrey Jordon

Expert2022-01-31Added 2575 answers

Answer is given below (on video)