Notes

4.1 The Derivative of a Function


"The basic idea in calculating the slope of the tangent line at a point $P_0$ on the curve is to find the slope of the line through $P_0$ and another point $P$ on the curve, and then to take the limit of that slope as $P$ approaches $P_0$.  If that limit exists, it is defined to be the slope of the curve at $P_0$."

If the coordinates of point $P_0$ are $(x_0, y_0)$ and the coordinates of point $P$ are $(x, y)$, then the slope of the line passing through those points is

          $\dfrac {y-y_0}{x-x_0}$    or    $\dfrac {f(x)-f(x_0)}{x-x_0}$.

Let $f:D \rightarrow R$ with $x_0 \in D$ an accumulation point of $D$ and $x_0 \in D$ with $x \neq x_0$, define $T(x)= \dfrac {f(x)-f(x_0)}{x-x_0}$.

Let $f:D \rightarrow R$ with $x_0$ an accumulation point of $D$ and $x_0 \in D$.  For each $t \in R$ such that $x_0 + t \in D$ and $t \neq 0$, define $Q(t)= \dfrac {f(x_0 + t)-f(x_0)}{t}$. 

Prove that the two definitions above are equivalent.