understanding of total derivative

Question

Let $f(x_1, ..., x_n)$ be a function defined in a some area of the point $x^0 \in \mathbf R^n$ and let the functions of one variable $\phi_1(t), ..., \phi_n(t)$ are defied and differentiable around some point $t^0 \in \mathbf R$. We also know that $x_i^0=\phi_i(t^0)$ for $1 \le i \le n$.

Let's assume that $f(x)$ is differiable in $x^0$, and $\phi_i(t)$ in $t^0$.

Then the composite function $F(t)=f(\phi_1(t),...,\phi_n(t))$ is differentiable and

$$ F'(t)=\sum_{i=1}^{n}{\frac{\partial f}{\partial x_i}(x^0)\phi_i'(t^0)} $$

OK, so what is the intuition behind this? Why are we summing?

My understanding is that this is a dot product of the vector of he derivatives of the outer functions with the vector of the derivatives of the inner functions $(\phi_1(t), ..., \phi_n(t))$ so that the total derivate kind of gives how much the vector of the inner derivatives increases the vector of the partial derivatives (or how much they point in the same direction).

Answer 1

It shows you the following based on the fact that the given expression is a dot-product:

• imagine you run on a parametric curve on the surface given by $(x,f(x))$ through point $x^0$
• if you run through $(x^0,f(x^0))$ in the direction of the gradient of $f$ at $x^0$, then you run in the direction of maximum increase on that surface