Chapter 4.2, Problem 10E

(a)

To determine

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

Answer to Problem 10E

  $\frac{dy}{dx}=-\frac{x}{\sqrt{\left(9-x^2\right)}}$

Explanation of Solution

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

  $x^2+y^2=9$

Calculation: Convert the equation in terms of y ,

  $y=\sqrt{\left(9-x^2\right)}$ ................(1)

Using the derivation,

  $\begin{array}{l}\frac{d\left(x^2\right)}{dx}+\frac{d\left(y^2\right)}{dx}=\frac{d\left(9\right)}{dx}\\ 2x+2y\times \frac{dy}{dx}=0\\ \frac{dy}{dx}=-\frac{2x}{2y}\\ \frac{dy}{dx}=-\frac{x}{y}\end{array}$

Substitute y from equation (1),

  $\frac{dy}{dx}=-\frac{x}{\sqrt{\left(9-x^2\right)}}$

So the derivative of y with respect to x is $\frac{dy}{dx}=-\frac{x}{\sqrt{\left(9-x^2\right)}}$

(b)

To determine

To calculate: The slope of curve at (0, 3).

Answer to Problem 10E

The slope of curve at given point is $0$ .

Explanation of Solution

Calculation: Put the given point in the differentiated equation,

  $\begin{array}{l}\frac{dy}{dx}=-\frac{0}{\sqrt{\left(9-{\left(0\right)}^2\right)}}\\ \frac{dy}{dx}=0\end{array}$

The slope of curve at given point is $0$ .