## Calculus: Early Transcendentals (2nd Edition)

When we think of the instantaneous rate of change at a given point we want to know how much the function changes per unit length of the narrow interval containing $a$ in the limit when the length of that interval tends to zero. This is exactly given by the expression for the tangent slope.
The slope of the tangent at point $(a,f(a))$ is given by $$m_{tan}=\lim_{x\to a}\frac{f(x)-f(a)}{x-a}$$. When we think of the instantaneous rate of change at a given point we want to know how much the function changes per unit length of the narrow interval containing $a$ in the limit when the length of that interval tends to zero. This is exactly given by the expression for the tangent slope.