chain rule application to multivariate derivative

Question

I have a book with this in it:

total derivative

But I don't understand where the single quote comes from.

If I look at this: http://tutorial.math.lamar.edu/Classes/CalcIII/ChainRule.aspx

I would assume the total derivative to be:

My take:

with dx/dt being x-dot and dt/dt reducing to 1.

But then I don't have a single quote, which indicates the partial of S with respect to x needs to differentiated with respect to time.

Could someone break down what i'm missing here?

Answer 1

It's hard to tell without the context (“I have a book” is useless information – tell us which book it is!), but my guess is that the prime means matrix transposition (like in Matlab, for instance), and that they view the gradient $\partial S/\partial \mathbf{x}$ as a column vector which must be turned into a row vector in order to be multiplied by the column vector $\dot{\mathbf{x}}$.