Question

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

Answer 1

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:

$f(x) = x^2$

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

$f'(x)=2x$

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:

$f'(5)=2(5)$

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