Q. 46 - 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.

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

Step-by-Step Solution

Answer

The set function is${\mathrm{S}}_1=\left\{\right(\mathrm{x},\mathrm{y})\mid \mathrm{x}\ne 0,\mathrm{y}\ne 0\}$for all points.

Step 1: Partial derivatives of function

Given function is,

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

So the partial derivatives of function are,

$\frac{\mathrm{d}}{\mathrm{dx}}\mathrm{f}(\mathrm{x},\mathrm{y})=-\frac{{\mathrm{x}}^2-{\mathrm{y}}^2}{{\left({\mathrm{x}}^2+{\mathrm{y}}^2\right)}^2}$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$\mathrm{x}\ne 0$and $\mathrm{y}\ne 0$, these partial derivatives are continuous.

They are continuous on the set${\mathrm{S}}_1=\left\{\right(\mathrm{x},\mathrm{y})\mid \mathrm{x}\ne 0,\mathrm{y}\ne 0\}$and differentiable at all points in ${\mathrm{S}}_1$.