Suppose $f:R^2\to R$ is differentiable and $F(x)=f(x,-x)$. I have tried to compute the derivative through 2 methods and had a sign problem. Could someone please point out where I messed up? The derivative is supposed to be unique!
Method 1: Chain rule:
${dg\over dx}={\partial f\over\partial x}{d x\over d x}+{\partial f\over\partial y}{d y\over d x}={\partial f\over\partial x}-{\partial f\over\partial y}$
Method 2: Definition:
$f(x+t,y+t)-f(x,y)-tL(x,y)$ $=f(x+t,y+t)-f(x+t,y)+f(x+t,y)-f(x,y)-tL(x,y)$ $=t{\partial f\over\partial y}+t{\partial f\over\partial x}+O(t^2)-tL(x,y)$
So it seems like I should take $L(x,y)={\partial f\over\partial y}+ {\partial f\over\partial x}$
Why is there a sign error?
Thanks