Chapter 4.2, Problem 29E

To determine

To Compute: Second order derivative of the given function.

Answer to Problem 29E

  $\frac{y^2-{(x+1)}^2}{y^3}$ .

Explanation of Solution

Given information: Given $y^2=x^2+2x$ .

Calculation:

  $y^2=x^2+2x$

Differentiate with respect to x,

  $\begin{array}{l}\frac{d(y^2)}{dy}\times \frac{dy}{dx}=\frac{d(x^2)}{dx}+2\times \frac{d(x)}{dx}\\ 2y\times \frac{dy}{dx}=2x+2\times 1\\ \frac{dy}{dx}=\frac{2(x+1)}{2y}\\ \frac{dy}{dx}=\frac{x+1}{y}\end{array}$

Again, differentiate with respect to x,

  $\begin{array}{l}\frac{d^{2}y}{d{x}^2}=\frac{\frac{d(x+1)}{dx}\times y-(x+1)\times \frac{dy}{dx}}{y^2}\\ \frac{d^{2}y}{d{x}^2}=\frac{1\times y-(x+1)\times \frac{(x+1)}{y}}{y^2}\\ \frac{d^{2}y}{d{x}^2}=\frac{y^2-{(x+1)}^2}{y^3}\end{array}$