my professor posted a solution to a partial derivative in his class notes. I am not sure how he got to <0,1,5>. Any suggestions would be greatly appreciated!
Let f(x, y) = xy + y2. Find the tangent lines to the curves obtained by slicing f(x,y) using the planes x = 1,y = 2 at (1,2). When we slice f(x,y) at (1,2) with x = 1, what we are doing is finding the tangent line to the curve f(1,y) at (1,2). We see that fy(x,y) = x+2y, so fy(1,2) = 5. This line passes through (1,2,6), and has direction vector given by ⟨0, 1, 5⟩. Indeed, we see that the x-coordinate of this line never changes, which explains why the first coordinate equals 0. This means that the tangent line in question has equation x = 1, y = 2 + t, z = 6 + 5t.