Can someone explain this to me : ($f$ is a differentiable function) why is $\frac{d}{dt} f(2,t) = ∇f(2,t) \cdot (0,1)$?
This reminds me of the formula of directional derivate but it's not the same.
I also would like to see a demonstration of this formula (I didn't know it before).
Let $g(t)=(2,t)$. Then $\frac{\mathrm d}{\mathrm dt}f(2,t)=\frac{\mathrm d}{\mathrm dt}(f\circ g)(t)$. But then you can apply the chain rule:$$\frac{\mathrm d}{\mathrm dt}(f\circ g)(t)=f'\bigl(g(t)\bigr)\bigl(g'(t)\bigr).$$But $g'(t)=(0,1)$ and so$$f'\bigl(g(t)\bigr)\bigl(g'(t)\bigr)=f'\bigl((2,t)\bigr)\bigl((0,1)\bigr)=\nabla f(2,t).(0,1).$$