Question

What is the difference between the slope of a tangent and instantaneous rate of change of `f(x)`?

Answer 1

There is no difference in the mathematical expressions for these things. The difference is interpretation (and setting).

The slope of a tangent line is a geometrical idea. It is tied to curves in a coordinate plane.

The instantaneous rate of change is a relationship between two variable quantities, one depending on the other.

The calculations or the algebra, if you like, are the same for both.

Consider: `(f(x+h)-f(x))/h`. What is it?

In terms of the symbols we write mathematically, there is no difference between the slope of a secant line, (geometry) the average rate of change (related variable quantities) and the difference quotient (an algebraic expression). But the mathematics has different meanings in different settings.

In a similar way (in the same way?): `lim_(hrarr0)(f(x+h)-f(x))/h` can be interpreted differently in different settings.