Answer
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 $(a,f(a))$.
Work Step by Step
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 $(a,f(a))$.
