What is the slope of the tangent line of (x-y)(x+y)-xy+y= C (xy)(x+y)xy+y=C, where C is an arbitrary constant, at (1,2)(1,2)?

1 Answer
Feb 2, 2016

The slope of the tangent is 00 (and CC is not "arbitrary", it is -33).

Explanation:

(x-y)(x+y)-xy+y= C (xy)(x+y)xy+y=C

Instead of differentiating immediately, let's rewrite the first product.

x^2-y^2-xy+y=Cx2y2xy+y=C.

This form is (I think) simpler to differentiate.

2x-2y dy/dx - y - x dy/dx + dy/dx = 02x2ydydxyxdydx+dydx=0

At the point (1,2)(1,2), we get

2(1) - 2(2) dy/dx - 2 - (1) dy/dx + dy/dx = 02(1)2(2)dydx2(1)dydx+dydx=0

Solve for dy/dxdydx to see that dy/dx = 0dydx=0