I'm trying to understand how the two relate from this definition
From my understanding, this definition of $D_{j} f(\vec a)$ is expanded as
$$D_{j} f(\vec a) = lim \frac{(f(a_{1}+te_{j_{1}} , .... , a_{m} + te_{j_{m}}) - f(a_{1} , ... a_{m}))}{t}$$
No special cases, I'm treating $\vec e_{j}$ as if it has non zero components,
Now with the definition of $ \phi(t) = f(a_{1} , ... , a_{j-1} , t , a_{j+1} , ... , a_{m})$
The ordinary derivative of $\phi(t)$ is:
$$\phi'(t) = lim \frac{\phi(t+h) - \phi(t)}{h}$$
Using the definition of $\phi(t)$ , it is expanded as:
$$\phi'(t) = lim \frac{ f(a_{1} , ... , a_{j-1} , t+h , a_{j+1} , ... , a_{m}) - f(a_{1} , ... , a_{j-1} , t , a_{j+1} , ... , a_{m})}{h}$$
and at $t=a_{j}$ it becomes:
$$\phi'(t) = lim \frac{ f(a_{1} , ... , a_{j-1} , a_{j}+h , a_{j+1} , ... , a_{m}) - f(a_{1} , ... , a_{j-1} , a_{j} , a_{j+1} , ... , a_{m})}{h}$$
So how are $\phi'(a_{j})$ and $D_{j}f(\vec a)$ the same? I can't see the connection.
Also I believe the parameter $t$ in $\phi$ is not the same as the $t$ in the limit of $D_{j}f(\vec a)$