How can y=(x-1)^2+2(x-1)(x+2)+(x+2)^2 be differentiated in two different ways?

Either we can differentiate each term of the sum, or we
can notice that the expression is a complete square and we'll differentiate the complete
square.

If we'll put (x-1) as a and (x+2) as b, and we'll
re-write the expresison, we'll get a perfect square:

y=a^2
+ 2ab + b^2

y = (a+b)^2

y =
(x-1+x+2)^2

We'll combine like
terms:

y = (2x+1)^2

Now, we'll
differentiate both sides, with respect to x, using chain
rule:

dy/dx =
2(2x+1)*(2x+1)'

dy/dx =
2(2x+1)*2

dy/dx =
4(2x+1)

We'll remove the
brackets:

dy/dx = 8x + 4

The
other method is to differentiate each term of the sum, with respect to
x.

dy/dx = d(x-1)^2/dx + 2d[(x-1)(x+2)]/dx +
d(x+2)^2/dx

dy/dx = 2(x-1)+ 2d(x^2+x-2)/dx +
2(x+2)

dy/dx = 2x - 2 + 2(2x+1) + 2x +
4

dy/dx = 4x + 2 + 4x +
2

dy/dx = 8x +
4

Both methods will lead to the same result:
dy/dx = 8x + 4.