Chapter 4.2, Problem 13E

To determine

The slope of the given curve.

Answer to Problem 13E

Slope is defined for all values of $x\text{and}y$ except when $x=0\text{and}x=2y$ .

Explanation of Solution

Given information: Given curve $x^{2}y-x{y}^2=4$ .

Formula used: $\frac{d(x^n)}{dx}=n{x}^{n-1}\text{and}\frac{d(\text{constant})}{dx}=0$ with product, chain and subtraction rule.

Calculation:

Let the intersection points of the curve and tangent is $P(x,y)$ .

Also, the slope of tangent at given points,

  $\begin{array}{l}{x}^{2}y-x{y}^2=4\\ \text{Differentiate}\text{with}\text{respect}\text{to}x\\ x^2\times \frac{dy}{dx}+\frac{d(x^2)}{dx}\times y-\left(x\times \frac{d(y^2)}{dy}\times \frac{dy}{dx}+\frac{d(x)}{dx}\times y^2\right)=\frac{d(4)}{dx}\\ x^2\times \frac{dy}{dx}+2x\times y-x\times 2y\times \frac{dy}{dx}-1\times y^2=0\\ x^2\frac{dy}{dx}-2xy\frac{dy}{dx}=y^2-2xy\\ (x^2-2xy)\frac{dy}{dx}=y^2-2xy\\ {\frac{dy}{dx}}_{(x,y)}=\frac{y(y-2x)}{x(x-2y)}\end{array}$

Now, the slope is not defined, when,

  $\begin{array}{l}x(x-2y)=0\\ x=0,x=2y\end{array}$

Thus, slope is defined for all values of $x\text{and}y$ except when $x=0\text{and}x=2y$ .