Gradient of a Mutivariable function

Question

I am having a multivariable function g(x,y) as below: How do I calculate the gradient of

$g(x, y) = 1/2 (x + y − |x − y|)$.

Answer 1

$$g(x,y)=\frac{x}{2}+\frac{y}{2}−\frac{|x−y|}{2}$$ $$\frac{\partial}{\partial x} g(x,y)= \frac{\partial}{\partial x} \left( \frac{x}{2} \right)+\frac{\partial}{\partial x} \left(\frac{y}{2} \right)−\frac{\partial}{\partial x} \left( \frac{|x−y|}{2} \right)$$ $$\frac{\partial}{\partial x} g(x,y)= \frac{1}{2}−\frac{1}{2}\frac{\partial}{\partial x} \left( |x−y| \right)$$ And by the chain rule: $$\frac{\partial}{\partial x} \left( |x−y| \right) = \left(\frac{d}{du} |u| \right)_{u=x-y}.\frac{\partial u}{\partial x}$$ And: $$\frac{d}{du} |u|=\frac{u}{|u|}$$ Putting it al together: $$\frac{\partial}{\partial x} g(x,y)= \frac{1}{2}−\frac{1}{2}\frac{x-y}{|x-y|} $$ Try the partial derivative to $y$. At the end: $$\nabla g(x,y) = \frac{\partial}{\partial x} g(x,y) \hat{i}+\frac{\partial}{\partial y} g(x,y) \hat{j}$$