the curvy d vs. d

Today in class we saw how confusing and intimidating the partial derivative can look when used as a result of the chain rule.

A recap of the difference between the derivative and the partial derivative.:

If w is a function of x and y, and x and y are functions of t, then w is essentially a function if t. That is why we can find the derivative of w with respect to t.
Using the chain rule, we arrive at this formula:

However, if w is a function of x and y, and x and y are functions of s and t, we can no longer find the derivative of w with respect of t. We can find the partial derivative of w with respect to t and s.