3.4 Definition of the derivative

The derivative measures the rate of change of a real function $f(x)$ with respect to $x$. Consider points $A=(a,f(a))$ and $B=(a+h,f(a+h))$ on its graph and draw the chord from $A$ to $B$. Then

This is called the difference quotient. Suppose that $A$ is fixed and let $B$ approach $A$ by letting $h\rightarrow 0$. Then $AB$ becomes tangent to the graph at $A$, if this tangent exists. Thus we have

where this limit exists. The value of the limit defines the derivative of $f$ at $a$, written $f^{\prime}(a)$ or ${{df}\over{dx}}(a).$