Can derivatives be used to measure the the slope of the tangent line at x=a

Yes. In fact, that is basically the definition of a derivative. It is the instantaneous rate of change of a function. 

For example, picture the graph of the following function:



The slope is constantly changing at every x-value, so to find  the slope at x=a, we find the  derivative of the  function.



Once we have the derivative, simply plug in a for x to find the slope of the line tangent to f(x) at x=a.

For example, at x=5:



The slope of f(x) at x=5 is 10.