Q. 45 - Calculus

Question

Find all points where the first-order partial derivatives of the functions in Exercises $43-54$ are continuous. Then use Theorems $12.28$ and $12.31$ to determine the sets in which the functions are differentiable.

$\mathrm{f}(\mathrm{x}, \mathrm{y})=\frac{\mathrm{x}}{{\mathrm{x}}^2+{\mathrm{y}}^2-1}$

Step-by-Step Solution

Answer

The sets is ${\mathrm{S}}_1=\left\{(\mathrm{x},\mathrm{y})\mid {\mathrm{x}}^2+{\mathrm{y}}^2\ne 1\right\}$and differentiable with ${\mathrm{S}}_1$all points.

Step 1: Partial derivatives of function

Given function is,

$\mathrm{f}(\mathrm{x},\mathrm{y})=\frac{\mathrm{x}}{{\mathrm{x}}^2+{\mathrm{y}}^2-1}$

So the partial derivatives of function are,

$\frac{\mathrm{d}}{\mathrm{dx}}\mathrm{f}(\mathrm{x},\mathrm{y})=-\frac{{\mathrm{x}}^2-{\mathrm{y}}^2+1}{{\left({\mathrm{x}}^2+{\mathrm{y}}^2-1\right)}^2}$And$\frac{\mathrm{d}}{\mathrm{dy}}\mathrm{f}(\mathrm{x},\mathrm{y})=-\frac{2\mathrm{xy}}{{\left({\mathrm{x}}^2+{\mathrm{y}}^2-1\right)}^2}$

Step 2: Explanation

Where ${\mathrm{x}}^2+{\mathrm{y}}^2\ne 1$, these partial derivatives are continuous.

They're continuous on the set${\mathrm{S}}_1=\left\{(\mathrm{x},\mathrm{y})\mid {\mathrm{x}}^2+{\mathrm{y}}^2\ne 1\right\}$ ,

And the function is differentiable at all points in${\mathrm{S}}_1$.