Calculate derivative of function y=2x^2-10x+13 +2(x^2-5x+6) using 2 methods.

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.

We'll re-write the expression of the function to
emphasize the fact that it represents a complet
square.

y=2x^2-10x+13  +
2(x^2-5x+6)

y = (x^2 - 4x + 4) + (x^2 - 6x + 9) +
2(x^2-5x+6)

y = (x-2)^2 + (x-3)^2 +
2(x^2-5x+6)

If we'll put (x-2) as a and (x-3) 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-2+x-3)^2

We'll
combine like terms:

y =
(2x-5)^2

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

dy/dx =
2(2x-5)*(2x-5)'

dy/dx =
2(2x-5)*2

dy/dx =
4(2x-5)

We'll remove the
brackets:

dy/dx = 8x - 20

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

dy/dx = d(x-2)^2/dx + 2d[(x-2)(x-3)]/dx +
d(x-3)^2/dx

dy/dx = 2(x-2)+ 2d(x^2-5x+6)/dx +
2(x-3)

dy/dx = 2x - 4 + 2(2x-5) + 2x
-6

dy/dx = 4x -10 + 4x
-10

dy/dx = 8x -
20

Both methods yields the same result: dy/dx
= 8x - 20.