# How to visualize and understand the derivative of a multivariable function along a path?

The theorem states that the derivative of f: $$R^n->R$$ along the path x̃ at $$t_0$$ is the derivative of f(x̃ (.)) at $$t_0$$, if it exists: $$df(x̃(t))/dt|_{t=t_0}= lim_{n->∞} f(x̃(t_0)) - f(x̃(t_n))/ t_0 - t_n$$

I am having trouble understanding and visualizing this. What exactly is this path and how is that related to two or more variables? What is x̃? I tried to find tutorials online, but these concepts were not explained. It's my first time studying this so I do not have a good background. Thanks for any help!

The idea is that a path in a multidimensional space can be described in terms of some parameter $$t$$. So, in $$\mathbb{R}^n$$, we can describe a path as $$\tilde{x}(t):\mathbb{R}\to\mathbb{R}^n$$. Basically, for every 'point in time' $$t$$, we fix some point in space $$\tilde{x}(t)$$.
Now, we have $$f:\mathbb{R}^n\to\mathbb{R}$$, basically, some number being assigned to each point in space (say, 'temperature' or something). We are interested in how this quantity changes as we move in space. But to be precise, we are moving along a path $$\tilde{x}(t)$$ in space. The simplest way to find the derivative is to fall back on the definition: If I move a little bit, how much does $$f$$ change?
So right now I'm starting at 'time' $$t_0$$, sitting at the point in space $$\tilde{x}(t_0)$$, and seeing a 'temperature' of $$f(\tilde{x}(t_0))$$. A little time later, say $$t_0+\varepsilon$$, I am at $$\tilde{x}(t_0+\varepsilon)$$, seeing a temperature of $$f(\tilde{x}(t_0+\varepsilon))$$. The derivative is just the difference of these two, divided by the change in time: $$\mathrm{d}\,f(\tilde{x}(t))\,/\mathrm{d}t\left.\right|_{t=t_0}=\lim_{\varepsilon\to0}(f(\tilde{x}(t_0+\varepsilon))-f(\tilde{x}(t_0)))/\varepsilon$$.
In your book they may have written $$t_0+\varepsilon$$ as $$t_n$$ instead, so the denominator looks like $$t_n-t_0$$ or something. But that's just details, I hope you understand the big picture here! Feel free to ask for clarification if something wasn't explained clearly.
• @alioshakaramazov $t$ is its own thing, independent of any spatial $x$ or $y$ coordinates. Commented Jan 9, 2023 at 21:50