Implicit differentiation with price and demand in single equation

Question

Price p and demand q are related by the equation

ln(pq+1)=100q

Find the rate of change of demand with respect to price when demand is 23:

??? items/dollar.

So, a unit increase in price will cause demand to fall by approximately how many items?

??? items.

Now what is the rate of change of revenue with respect to price when demand is 23?

??? items.

So, a unit increase in price will cause revenue to change by approximately how many dollars?

??? items.

Answer 1

ln(pq + 1) = 100q;

We do implicit differention by price p: 1/(pq + 1)(q + pdq/dp) = 100dq/dp;

(q + pdq/dp)/(pq + 1) = 100dq/dp; q/(pq + 1) = (100 - p/(pq + 1))dq/dp;

dq/dp = q/(100pq + 100 - p).

If q = 23, then ln(23p + 1) = 2300; 23p + 1 = e²³⁰⁰. Somthing wrong now with data, because e²³⁰⁰ _is ^{astronomically huge}