Why the slope of the tangent line can be interpreted as an instantaneous rate of change?

How do we define the derivative of a function? First, identify two points on a given curve: (x, f(x)) and ((x+h), f(x+h)), and write out the rate of change expression:

f(x+h) - f(x) f(x+h) - f(x)

=

x+h-x h

If done properly, your expression for f(x+h)-f(x) will be evenly divisible by h. Obtain this quotient. Last, let h decrease towards zero. The limit as h approaches zero is the instantaneous rate of change of the given function.

The line connecting (x,f(x)) and (x+h, f(x+h) is initially a secant line:

f(x+h) - f(x) f(x+h) - f(x)

=

x+h-x h

As we take the limit of this secant line expression, the secant line becomes the tangent line to the curve, also referred to as the "derivative" or "instantaneous rate of change" of the function with respect to x.