Why is it better to have the private key as the multiplicative inverse mod phi(n) in RSA? I'm studying the RSA algorithm and the theory says to select $d$ to be the multiplicative inverse of. 

$$
e \mod \phi(n) 
$$

If we take this approach I know that it is important that $e$ and  $\phi(n)$ are coprime.

Why is this "better" as opposed to say the following equation where $e$ is still selected as a coprime to $\phi(n)$:

$$
 d = \frac{\phi(n) + 1}{e}
$$

I would like to have a deep explanation. I do not have formal math training, so any links to help understand the answer is very much appreciated.
In the standarized RSA algorithm the private key $d$ is calculated computing the modular multiplicative inverse with the Extended Euclidean GCD that satisfies:

$1\equiv e \cdot d\pmod {\varphi(p \cdot q})$

Notice that modular multiplicative inverse can be expressed as:

$$d=\frac{\varphi(p\cdot q)\cdot k + 1}{e}$$

for some $k$ multiple of $\varphi$($p\cdot q$). We know that $e$ is 17 bits (65537) so $k$ will be small, thus computing $d$ with this method would be realizable. You have to know the factorization of the semiprime for computing $\varphi(p \cdot q)$.

Mathematically other ways for computing the private key exist, such as the Euler Criterion.

For calculating the modular multiplicative inverse we would have:

$d\equiv e^{\varphi(\varphi(p \cdot q))-1}\pmod {\varphi(p \cdot q})$

As you can see we would need to have the factorization of $\varphi(p\cdot q)$, so we cannot deal with big semiprime modulus. Concretely we need the factors of $(p-1)$ and $(q-1)$ in order to compute $\varphi(p-1) \cdot \varphi(q-1)$