# Find the rate of change of demand (x) of a commodity with respect to its price (y) if y = 5x+72x-13 - Mathematics and Statistics

Sum

Find the rate of change of demand (x) of a commodity with respect to its price (y) if y = (5x + 7)/(2x - 13)

#### Solution

y = (5x + 7)/(2x - 13)

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

("d"y)/("d"x) = "d"/("d"x) ((5x + 7)/(2x - 13))

= ((2x - 13)*"d"/("d"x) (5x + 7) - (5x + 7)*"d"/("d"x)(2x - 13))/(2x - 13)^2

= ((2x - 13)(5 xx 1 + 0) - (5x + 7)(2 xx 1 - 0))/(2x - 13)^2

= ((2x - 13)(5) - (5x + 7)(2))/(2x - 13)^2

= (10x - 65 - 10x - 14)/(2x - 13)^2

∴ ("d"y)/("d"x) = (-79)/(2x - 13)^2

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

("d"x)/("d"y) = 1/((("d"y)/("d"x))), where "dy"/"dx" ne 0

i.e. ("d"x)/("d"y) = 1/((- 79)/(2x - 13)^2)

= (-(2x - 13)^2)/79

Concept: Derivatives of Composite Functions - Chain Rule
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