Having watched an integralCALC video lesson, given
$$w=xe^{y/z}, x = t^2, y = 1-t, z=1+2t$$ which could be rewritten as $$w=t^2 e^\frac{1-t}{1+2t}$$
How does $dw/dt$ differ from $\partial w/\partial t$? Is there some intuition behind the partial derivative (such as a geometric interpretation)?
There's no difference. It's just that we use straight $d$s to denote derivatives of functions with respect to one variable. It may seem like the function $w$ depends on more than one variable, but the other variables all depend on $t$, so at the end of the day all the values of the function $w$ are determined by the values of $t$, so we use the straight $d$s to denote its derivative.
There’s no difference in this case, but there could be a situation where there were a difference.
Suppose that instead of the original problem, we had written $w=t^2e^{y/z}$, $y=1-t$, and $z=1+2t$. Since this is the same $w$, $dw/dt$ is the same. But if someone were to write $w(t,y,z)=t^2e^{y/z}$, then they could decide that $\partial w/\partial t$ should mean to differentiate only with respect to the first variable: $\partial w/\partial t=2te^{y/z}=2te^{\frac{1-t}{1+2t}}$. However, that someone should clearly indicate that is what they mean by saying, for example, “where $\partial w/\partial t$ is differentiating only with respect to the first variable in $w(t,y,z)$”.
