Usually what's meant by the symbols is
It's important to keep in mind that the derivative with respect to contravariant vector components (upper indices) gives gives an operator that transforms like a covariant component (lower indices). This is, of course, only true in flat space-time!
You have to use the product rule for derivatives. I would also like to suggest that since the alphas in the first factor are dummy variables, you change those two to ##\gamma## before you begin taking the derivative with respect to ##\phi^\alpha##. Either that, or write out exactly what the notation ##\partial_\alpha\phi^\alpha\partial_\beta\phi^\beta## means before you start.thanks vanhees71
I know that, but i ask can me consider these vectors as two multiplied functions (quadratic ∂ ø) and derivative becomes 2 ∂αø?
You have to use the product rule for derivatives. I would also like to suggest that since the alphas in the first factor are dummy variables, you change those two to ##\gamma## before you begin taking the derivative with respect to ##\phi^\alpha##. Either that, or write out exactly what the notation ##\partial_\alpha\phi^\alpha\partial_\beta\phi^\beta## means before you start.