Q. 43 - Calculus

Question

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

$f(x,y)=x^2-y^2$

Step-by-Step Solution

Answer

These partial derivatives are continuous at all points. They are continuous on the set $S_1=\left\{\right(x,y)\mid x,y\in 9\}$ and $f$differentiable at every point in $S_1$

Step 1: Expression of solution

We have given $f(x,y)=x^2-y^2$

The partial derivatives of function are 

$\frac{d}{dx}f(x,y)=2x \text{ and }\frac{d}{dy}f(x,y)=-2x$

Step 2: Continuous Set

They are continuous on the set  $S_1=\left\{\right(x,y)\mid x,y\in 9\}$ and $f$ is differentiable at every point in $S_1$