Tutorial: The derivative: algebraic viewpoint
(This topic is also in Section 3.6 in Applied Calculus or Section 10.6 in Finite Mathematics and Applied Calculus)) #[I don't like this new tutorial. Take me back to the old tutorial!][No me gusta este nueve tutorial. ¡Regresame al tutorial más viejo!]#
Calculating the derivative at a point algebraically
%%Q So far, all we have been doing is approximating the derivative of a function using numerical and graphical approaches. Is there a way of computing it exactly?%%A When the function is specified algebraically, we can calculate the exact value of the derivative using an algebraic approach, which we show here. The algebraic approach is quite straightforward: Instead of subtracting numbers to estimate the average rate of change over smaller and smaller intervals, we subtract algebraic expressions using the definition of the derivative at a point $x = a$ in terms of the difference quotient

$\displaystyle f'(a) = \lim_{h \to 0} \frac{f(a + h)  f(a)}{h} \qquad$ The derivative is the limit of the difference quotient.
$\displaystyle f'(x)$ \t $\displaystyle {}= \lim_{h \to 0} \frac{f(x + h)  f(x)}{h} \qquad$ \t The function $f'$ assigns to each $x$ the value $f'(x)$ of the derivative of $f$ at $x$.
\\ \t !2! ${}={}$Slope of tangent at the point $(x, f(x))$ on the graph, as illustrated here:
What we have calculated above is the value of the derivative function $f'$ at a particular $x$. But we can also use the same technique to give us a formula for $f'(x)$ by calculating it at a general $x$:
An application: Calculating exact velocity
We saw in the %%prevsectut that, if $f$ is a function of time $t$ (such as, for instance, the height of a launch vehicle) then its derivative $f'$ gives the velocity of $f$ (for instance the velocity of the launch vehicle).
Calculating the derivative of a rational function
Now let's repeat the method above with a function other than a quadratic function.
Now try the exercises in Section 3.6 in Applied Calculus or Section 10.6 in Finite Mathematics and Applied Calculus).
Copyright © 2018 Stefan Waner and Steven R. Costenoble