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

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You cannot ask more than one question at a time. So I am only providing one basic method.

We need to find the derivative of the function y=2x^2-10x+13 +2(x^2-5x+6).

y= 2x^2-10x+13 +2(x^2-5x+6)

=> y = 2x^2 - 10x + 13 + 2x^2 - 10x + 12

=> y = 4x^2 - 20x + 25

If f(x) = x^n, f'(x) = n*x^(n - 1)

y' = 2*4x - 20

=> y' = 8x - 20

**The required derivative of the function is y' = 8x - 20**

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.**