Linear Approximation: Find the linear approximation at a point Question:
f(x,y)=√(7+2xy)
Find the linear approximation at (3, -1)
My answer:
So I took the partial derivative at x and y, and I got Fx = y/(√(7+2xy)) and Fy = x/(√(7+2xy)). Evaluating for this, I get Fx(3,-1)= -1 and Fy(3,-1) = 3. Then L(x)= -x + 3y + 1, but I'm wrong. What did I do wrong?

make a change of variable $x = 3+h, y = -1 + k.$ then $$z = \left(7+ 2(3+h)(-1+k)\right)^{1/2} = \left(1-2h+6k+\cdots\right)^{1/2}= 1 + \frac 12(-2h+6k)+\cdots = 1-h+3k+\cdots$$
the planar approximation of $z$ at $(3, -1)$ is $z = 1 -(x-3) + 3(y+1).$
