What is the difference between $d$ and $\partial$? After seeing the following equation in a lecture about tensor analysis, I became confused.
$$
\frac{d\phi}{ds}=\frac{\partial \phi}{\partial x^m}\frac{dx^m}{ds}
$$
What exactly is the difference between $d$ and $\partial$?
 A: The difference is whether the rest of the variables of $f$ are considered constants  or variables  in $x.$ Former is partial, latter is total.


*

*$\frac{\partial f}{\partial x}$ is the partial derivative: $f$ is differentiated w.r.t. to $x$ while all other variables are considered constants in $x.$

*$\frac{d f}{d x}$ the is total derivative: $f$ is differentiated w.r.t. to $x$ while nothing is assumed about the other variables; they are considered variables in $x.$ (some variables might be, in fact, constants in $x.$)
A: As mentioned $d$ means total and $\partial$ partial derivative and are not the same. Total derivative also counts $x$ dependencies in other variables. For instance:
$$
f(x,v) = x^2 + v(x) \\
\frac{\partial f}{\partial x} = 2x \\
\frac{\partial f}{\partial v} = 1 \\
\frac{d f}{d x} = 2x + \frac{\partial v(x)}{\partial x} 
$$
Your formula most probably uses Einstein notation and is only a shorter way to write
$$
\frac{d\phi}{ds}= \sum_m \frac{\partial \phi}{\partial x^m}\frac{dx^m}{ds}
$$
