# Thread: The meaning of a derivative

1. ## The meaning of a derivative

I need some help understanding what exactly the derivative means in this problem:

The number $N$ of gallons of regular unleaded gasoline sold be a gas station at a price of $p$ dollars per gallon is given by $N=f(p)$.

What would $f'(2.959)$ mean? Would this value usually be positive or negative and why?

2. Originally Posted by hemi
I need some help understanding what exactly the derivative means in this problem:

The number $N$ of gallons of regular unleaded gasoline sold be a gas station at a price of $p$ dollars per gallon is given by $N=f(p)$.

What would $f'(2.959)$ mean? Would this value usually be positive or negative and why?
$f'(2.959)$ represents an instantaneous rate of change at the moment that $p = 2.959$. An average rate fof change would be going from say 2.959 to say 3.00 and then we would calculate

$
\frac{f(3)-f(2.959)}{3-2.959}
$

and further

$
\frac{f(3)-f(2.959)}{3-2.959} \approx f'(2.959)
$

3. This derivative might answer the question: "If you raise the price of gas, how much more (positive) or less (negative) gas will you sell?" There is no information about positive or negative.

4. Well, I, at least, tend to drive less and would consider buying a car with better gas milage as the price of gas goes up! That means that the amount of gas bought or sold would tend to decrease as the price goes up and so the derivative would be negative.