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3-59.

MORE NOTABLE NOTATION FOR THE DERIVATIVE

The use of $\frac { d y } { d x }$ comes from $\frac { \Delta y } { \Delta x }$, which is an expression for slope read as “the change in $y$ over the change in $x$”. We use $\Delta$ to represent change. When the change gets smaller and smaller until it is infinitely small (infinitesimal) we use the symbol $d$.

It is useful to think of change when working with derivatives. For example $\frac { d h } { d t }$ can represent the change in the height of an object with respect to time. Create expressions using the symbol $d$ that represent the following instantaneous change statements.

1. The change in the velocity, $v$, with respect to time.

instantaneous acceleration  $\frac{d?}{dt}$

2. The change in volume, $V$, with respect to the radius, $r$, of a cone.

$\frac{dV}{dr}$

3. The change in area, $A$, of a circle with respect to the perimeter, $p$.