Chapter 4.2, Problem 11E

(a)

To determine

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

Answer to Problem 11E

  $\frac{dy}{dx}=-\frac{x-1}{y-1}$

Explanation of Solution

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

  ${\left(x-1\right)}^2+{\left(y-1\right)}^2=13$

Calculation: Using the derivation,

  $\begin{array}{l}\frac{d\left({\left(x-1\right)}^2\right)}{dx}+\frac{d\left({\left(y-1\right)}^2\right)}{dx}=\frac{d\left(13\right)}{dx}\\ 2\left(x-1\right)\times \frac{d\left(x-1\right)}{dx}+2\left(y-1\right)\times \frac{d\left(y-1\right)}{dx}=0\\ 2\left(x-1\right)\times \left(1\right)+2\left(y-1\right)\times \frac{dy}{dx}=0\\ \frac{dy}{dx}=-\frac{2\left(x-1\right)}{2\left(y-1\right)}\\ \frac{dy}{dx}=-\frac{x-1}{y-1}\end{array}$

So the derivative of y with respect to x is $\frac{dy}{dx}=-\frac{x-1}{y-1}$

(b)

To determine

To calculate: The slope of curve at (3, 4).

Answer to Problem 11E

The slope of curve at given point is $-\frac{2}{3}$ .

Explanation of Solution

Calculation: Put the given point in the differentiated equation,

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

The slope of curve at given point is $-\frac{2}{3}$ .