I have a function $f: \mathbb{R}^{n} \rightarrow \mathbb{R}$
and a function $\phi: \mathbb{R}^{n-1} \rightarrow \mathbb{R}$
$f$ is defined as $f(x_1,...,x_n) = f(x_{1},...x_{i-1},\phi_{i}(x_{1},...,x_{i-1},x_{i+1},...,x_{n}),x_{i+1},...,x_{n})$ and differentiable.
for any $x_j\;$ such that $i\neq j\;$ why is $\frac{\partial f}{\partial x_{j}}=\frac{\partial f(x_{1},...x_{i-1},\phi_{i}(x_{1},...,x_{i-1},x_{i+1},...,x_{n}),x_{i+1},...,x_{n})}{\partial x_{j}}=\frac{\partial f}{\partial x_{j}}+\frac{\partial f}{\partial x_{i}}\cdot\frac{\partial\phi_{i}}{x_{j}}\;$
How is the chain rule applied here?