Proof for private derivatives

[n1nja_b0ss]

Let f(x,y) = |x + y|*sin|x + y|
a) Prove that f(x,y) has private derivatives at every point (x,y) in R²
b) Find the points where df/dx (x,y) and df/dy (x,y) are continuous
c) find where f(x,y) is differentiable

would appreciate help in any of the three points!

 

[Steven G]

Where are you stuck? What have you tried? Can we see your work? If you followed the guidelines you would have received help by now.

Hint: Remove the absolute bars.

 

[HallsofIvy]

"private derivatives"? Was that a mis-translation that should have been "partial derivatives"? As long as x+ y is not 0, so \(\displaystyle y\ne -x\), you can, as Jomo suggested, drop the absolute value bars. Since |x| is NOT differentiable at x= 0, you need to be more careful, perhaps using the limit definition of the derivative along the line y= -x.