help with derivation of contravarient transformation

Question

Note:this equation uses Einstein's notation The above equation is derivation of coordinate transformation rule for covariant vectors by applying chain rule on the formula for gradient..my question is how can we use a $x^{s}$ as its a summation of different variables..To illustrate lets say the vector is in ℝ³ therefore s=1,2,3(ie x,y,z ) if i expand the formula ill get $$\frac{∂x¹}{∂x*}\frac{∂∅}{∂x¹} +\frac{∂x²}{∂x*}\frac{∂∅}{∂x²}+\frac{∂x³}{∂x*}\frac{∂∅}{∂x³}$$ which on cancellation will give 3$\frac{∂∅}{∂x*}$≠$\frac{∂∅}{∂x*}$ do why can we so that?