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

Mar 25, 2015

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: $\frac{f \left(x + h\right) - f \left(x\right)}{h}$. 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?): ${\lim}_{h \rightarrow 0} \frac{f \left(x + h\right) - f \left(x\right)}{h}$ can be interpreted differently in different settings.