I see the elasticity value(s) of demand being used frequently to determine the relation between demand and variables such as price and income. I have however been unable to find any uses for the slope of the demand curve. Are there any?
Also, is there any relationship between the slope and the elasticity?
Price elasticity of demand is related (but not equal) to the inverse of the slope of the demand curve.
This Wikipedia article defines PED as:
Price elasticity of demand (PED or Ed) is a measure used in economics to show the responsiveness, or elasticity, of the quantity demanded of a good or service to a change in its price, ceteris paribus. More precisely, it gives the percentage change in quantity demanded in response to a one percent change in price. [emphasis added]
Look at the following (canonical) supply-demand graph.
Let's define $P^*$ the price of market equilibrium, $Q_D$ as the quantity demanded and $E_D$ as price elasticity of demand. Then:
$$ {slope} = \frac{rise}{run} = \frac{\Delta P}{\Delta Q_D} $$
And in mathematical terms, the verbal description of the PED reduces to:
$$ E_D = \frac{\frac{\Delta Q_D}{Q_D} \cdot 100}{\frac{\Delta P}{P^*} \cdot 100} $$
$$ E_D = \frac{P^*}{Q_D} \cdot \frac{\Delta Q_D} {\Delta P} $$
$$ E_D = \frac{P^*}{Q_D} \cdot \frac{1}{slope} $$
Constructive comments have been incorporated into this answer.
In particular, @Kontorus pointed out:
The PED is a percentage change, the slope of the demand curve is an absolute change.
And @denesp pointed out that:
... linear demand curves have constant slopes but changing elasticity. Cobb-Douglas derived demand (hyperbole) has changing slope but constant elasticity.
