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Question
Find the rate of change of demand (x) of a commodity with respect to its price (y) if y = 25x + log(1 + x2)
Solution
y = 25x + log(1 + x2)
Differentiating both sides w.r.t.x, we get
`"dy"/"dx" = "d"/"dx"[25"x" + log(1 + "x"^2)]`
`= "d"/"dx"(25"x") + "d"/"dx"[log(1 + "x"^2)]`
`= 25 + 1/(1 + "x"^2)*"d"/"dx"(1 + "x"^2)`
`= 25 + 1/(1 + "x"^2) * (0 + "2x")`
`= 25 + "2x"/(1 + "x"^2)`
`= (25(1 + "x"^2) + "2x")/(1 + "x"^2)`
∴ `"dy"/"dx" = (25 + 25"x"^2 + 2"x")/(1 + "x"^2)`
Now, by 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/((25 + 25"x"^2 + 2"x")/(1 + "x"^2)) = (1 + "x"^2)/(25"x"^2 + 2"x" + 25)`
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