## Calculus: Early Transcendentals (2nd Edition)

The instantaneous rate of change at $a$ is $$\lim_{x\to a}\frac{f(x)-f(a)}{x-a}.$$ The slope of the tangent at the given point $(a,f(a))$ is $$m_{tan}=\lim_{x\to a}\frac{f(x)-f(a)}{x-a}.$$ The value of the derivative of the function $f$ at the point $a$ is given by deffinition as $$f'(a)=\lim_{x\to a}\frac{f(x)-f(a)}{x-a}.$$ We see that at the given point of the graph the instantaneous rate of change, the slope of the tangent and the value of the derivative are equal to each other.