I'm reading about directional derivatives in $\mathbb{R}^n$. As an example, in $\mathbb{R}^3$ the directional derivative at $(x_0,y_0,z_0)$ in the direction of the unit vector $u=[a,b,c]^T$ is defined as
$$D_uf(x_0,y_0,z_0)=\lim_{t\to 0}\frac{f(x_0+ta,y_0+tb,z_0+tc)-f(x_0,y_0,z_0)}{t}$$
Then the following is written:
To remind ourselves of some other equivalent notations, notice that if $p=(x_0,y_0,z_0)$ and $u=[a,b,c]^T$, then we can also write $$\lim_{t\to 0}\frac{f(x_0+ta,y_0+tb,z_0+tc)-f(x_0,y_0,z_0)}{t}=\frac{d}{dt}\big(f(p+tu)\big)\bigg|_{t=0}$$
Why are these expressions equivalent? Would ideally like both an intuitive and rigorous clarification. My current line of thought is that
$$\frac{d}{dt}\big(f(p+tu)\big)\bigg|_{t=0}=\lim_{h\to0}\frac{f(p+(t+h)u)-f(p+tu)}{h}\bigg|_{t=0}=\lim_{h\to0}\frac{f(p+hu)-f(p)}{h}$$
So the roundabout way of defining it as the expression on the right could be because "a small increment" isn't just a small scalar, but a small scalar multiple of the vector $u$. I'm not sure about this line of reasoning though; am I on the right track?