I assume you mean using the basic definition of the derivative rather than the product rule.

No, what you have written is NOT correct. The derivative of a product is NOT the product of the derivatives. The derivative would be given by

The crucial point you need is that f(x)g(x)- f(a)g(a)= f(x)g(x)- f(a)g(x)+ f(a)g(x)- f(a)g(a)= (f(x)- f(a))g(x)+ f(a)(g(x)- g(a)).