Find the gradient of x2 + 2xy−2y2 + x = 2 at point (−4,1).

Yay, implicit differentiation

take derivitive of both sides

2x+2(y+xdy/dx)-4ydy/dx+1=0
solve for dy/dx
2x+2y+2x dy/dx-4y dy/dx+1=0
minus (2x+2y+1) from both sides
2x dy/dx-4y dy/dx=-2x-2y-1
dy/dx(2x-4y)=-2x-2y-1
divide both sides by (2x-4y)
dy/dx=(-2x-2y-1)/(2x-4y)
dy/dx=(2x+2y+1)/(4y-2x)

at (-4,1)
x=-4 and y=1

dy/dx=(2x+2y+1)/(4y-2x)
dy/dx=(2(-4)+2(1)+1)/(4(1)-2(-4))
dy/dx=(-8+2+1)/(4+8)
dy/dx=(-5)/(12)

the slope or gradient is -5/12

7for nonmembers i think