Question

Find the rate of change of demand (x) of a commodity with respect to its price (y) if y = 18x + log(x - 4).

Answer 1

y = 18x + log(x - 4)

Differentiating both sides w.r.t.x, we get

`"dy"/"dx" = "d"/"dx"`[18x + log(x - 4)]

`= "d"/"dx" (18"x") + "d"/"dx"`[log (x - 4)]

`= 18 + 1/("x - 4") * "d"/"dx"`(x - 4)

`= 18 + 1/("x - 4") * (1 - 0)`

`= 18 + 1/"x - 4"`

`= (18 ("x - 4") + 1)/("x - 4")`

`= (18"x" - 72 + 1)/("x - 4")`

∴ `"dy"/"dx" = (18"x" - 71)/("x - 4")`

Now, by a derivative of inverse function, the rate of change of demand (x) w.r.t. price (y) is

`"dx"/"dy" = 1/("dy"/"dx")`, where `"dy"/"dx" ne 0`.

i.e. `"dx"/"dy" = 1/((18"x" - 71)/("x - 4")) = ("x - 4")/(18"x" - 71)`