Chapter 4.2, Problem 12E

(a)

To determine

To calculate: The derivative of y with respect to x .

Answer to Problem 12E

  $\frac{dy}{dx}=-\frac{x+2}{y+3}$

Explanation of Solution

Given information: The equation of y in terms of x is given as,

  ${\left(x+2\right)}^2+{\left(y+3\right)}^2=25$

Calculation: Using the derivation,

  $\begin{array}{l}\frac{d\left({\left(x+2\right)}^2\right)}{dx}+\frac{d\left({\left(y+3\right)}^2\right)}{dx}=\frac{d\left(25\right)}{dx}\\ 2\left(x+2\right)\times \frac{d\left(x+2\right)}{dx}+2\left(y+3\right)\times \frac{d\left(y+3\right)}{dx}=0\\ 2\left(x+2\right)\times \left(1\right)+2\left(y+3\right)\times \frac{dy}{dx}=0\\ \frac{dy}{dx}=-\frac{2\left(x+2\right)}{2\left(y+3\right)}\\ \frac{dy}{dx}=-\frac{x+2}{y+3}\end{array}$

So the derivative of y with respect to x is $\frac{dy}{dx}=-\frac{x+2}{y+3}$

(b)

To determine

To calculate: The slope of curve at (1, -7).

Answer to Problem 12E

The slope of curve at given point is $\frac{3}{4}$ .

Explanation of Solution

Calculation: Put the given point in the differentiated equation,

  $\begin{array}{l}\frac{dy}{dx}=-\frac{x+2}{y+3}\\ \frac{dy}{dx}=-\frac{1+2}{-7+3}\\ \frac{dy}{dx}=-\frac{3}{-4}\\ \frac{dy}{dx}=\frac{3}{4}\end{array}$

The slope of curve at given point is $\frac{3}{4}$ .