The Derivative of a Function at a Point

To sum it all up we have the following definition, and method(s) to compute, the derivative of a function at a particular value:

Definition: The derivative of $f$ at a number $x=a$, denoted by $f'(a)$, is the value $$f'(a)=\lim_{x \to a} \frac{f(x)-f(a)}{x-a},
\text{ or equivalently,  }\quad  f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$$  if the limit exists.


The meaning of this derivative

The number which is the derivative of $f$ at $x=a$, denoted by $f'(a)$, gives us the instantaneous rate of change of the function $f$ with respect to $x$ when $x=a$.

If we are considering the graph of $f$, $f'(a)$ gives us the slope of the tangent line of $f$ at $x=a$.

DO:  Learn:  The derivative of a function at a point gives us the
                        instantaneous rate of change of that function at that point and the
                                slope of the tangent line of that function at that point.

Warning:  The definition of the derivative at a point is in the box above.  Some of the representations given by the derivative are in the paragraph immediately above, and these are not the definition.


Applications and units of derivatives

 If $f(t)$ represents the length of a deer's antler, in inches, on a given day in May, and $t$ represents time in days (with 1 being the first day of May), what does $f'(10)=.2$ represent?  Try to think this through before reading on, including units in your answer.

$f'(10)$ should represent the rate of change of $f$ on the day $t=10$, so it is the rate of change of the length of the deer's antler on May 10.  So if $f'(10)=.2$, the deer's antler is growing at a rate of .2 units on the 10th of May.  The units are determined by the change in f divided by the change in $t$, so the units are (units of $f$)/(units of $t$).  We see that the deer's antler is growing at a rate of .2 inches/day on the 10th of May.

Do:  Let $h(\theta)$ represent the height in meters of the end of a long ladder that is being raised, where $\theta$, measured in radians, represents the angle of the ladder to the (horizontal) ground.  What does $h'(\frac{\pi}{6})=3$ represent, including units?