## Calculus: Early Transcendentals 8th Edition

The definition is $$f'(a)=\lim_{x\to a}\frac{f(x)-f(a)}{x-a}.$$ The interpretations are that the derivative represents the instantaneous rate of change of the function and the slope of the tangent to the graph of the function at $(a,f(a))$.
The definition of the derivative is $$f'(a)=\lim_{x\to a}\frac{f(x)-f(a)}{x-a}.$$ The definition of the instantaneous rate of change of the function $f$ at $x=a$ is $$m_{inst}=\lim_{x\to a}\frac{f(x)-f(a)}{x-a}.$$ The definition of the slope of the tangent to the graph of the function $f$ at the point $(a,f(a))$ is given by $$m=\lim_{x\to a}\frac{f(x)-f(a)}{x-a}.$$ We see that these definitions are identical, so we see that the value of the derivative of the function $f$ at $a$ can be interpreted as: 1) the instantaneous rate of change of the function at $x=a$; 2) the slope of the tangent to the graph of the function at the point $(a,f(a))$.