Q. 48 - 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.

$\mathrm{f}(\mathrm{x},\mathrm{y})=\cos \left(\mathrm{xy}\right)$

Step-by-Step Solution

Answer

The set is ${\mathrm{S}}_1=\left\{\right(\mathrm{x},\mathrm{y})\mid \mathrm{x},\mathrm{y}\in \mathrm{ℝ}\}$for a function.

Step 1: Partial derivatives of function

Given function is,

$\mathrm{f}(\mathrm{x},\mathrm{y})=Cos\left(\mathrm{xy}\right)$

So the partial derivatives of function are,

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

Step 2: Explanation

At all places, these partial derivatives are continuous. 

They're continuous on the set ${\mathrm{S}}_1=\left\{\right(\mathrm{x},\mathrm{y})\mid \mathrm{x},\mathrm{y}\in \mathrm{ℝ}\}$, and the function is differentiable at all points in ${\mathrm{S}}_1$.