Cabiolab

2021-02-22

Second derivatives Find

Arnold Odonnell

Skilled2021-02-24Added 109 answers

Step 1

Given equation is$x+{y}^{2}=1$ .

To find second derivatives$\frac{{d}^{2}y}{{dx}^{2}}$ .

Solution:

Power rule of differentiation states that:

$\frac{d}{dx}\left({x}^{n}\right)=n{x}^{n-1}$

Chain rule of differentiation states that:

$\frac{d}{dx}\left[f\left(g\left(x\right)\right)\right]={f}^{\prime}\left(g\left(x\right)\right)\cdot {g}^{\prime}\left(x\right)$

Step 2

Differentiating the given function with respect to x,

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

$1+2y\frac{dy}{dx}=0$

$2y\frac{dy}{dx}=-1$

$\frac{dy}{dx}=-\frac{1}{2y}$

Again differentiating with respect to x,

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

$=-\frac{1}{2}\frac{d}{dx}\left(\frac{1}{y}\right)$

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

$=\frac{1}{2{y}^{2}}\cdot \frac{1}{-2y}$

$=-\frac{1}{4{y}^{3}}\text{}\text{}\text{}[U\mathrm{sin}g\frac{dy}{dx}=-\frac{1}{2y}]$

Step 3

Hence,$\frac{{d}^{2}y}{{dx}^{2}}=-\frac{1}{4{y}^{3}}$ .

Given equation is

To find second derivatives

Solution:

Power rule of differentiation states that:

Chain rule of differentiation states that:

Step 2

Differentiating the given function with respect to x,

Again differentiating with respect to x,

Step 3

Hence,