Advertisements
Advertisements
Question
Find the marginal demand of a commodity where demand is x and price is y.
y = `"x"*"e"^-"x" + 7`
Solution
y = `"x"*"e"^-"x" + 7`
Differentiating both sides w.r.t.x, we get
`"dy"/"dx" = "d"/"dx"("x"*"e"^-"x" + 7)`
`= "d"/"dx"("x"*"e"^-"x") + "d"/"dx" (7)`
`= "x" * "d"/"dx"("e"^-"x") + "e"^-"x" * "d"/"dx"("x") + 0`
`= "x" * "e"^-"x" * "d"/"dx"(- "x") + "e"^-"x"(1)`
`= "x" * "e"^-"x" (- 1) + "e"^-"x"`
`= "e"^-"x"(- "x" + 1)`
∴ `"dy"/"dx" = (- "x" + 1)/"e"^"x"`
Now, by derivative of inverse function, the marginal demand of a commodity is
`"dx"/"dy" = 1/("dy"/"dx")`, where `("dy"/"dx") ne 0`
i.e. `"dx"/"dy" = 1/((- "x" + 1)/"e"^"x") = "e"^"x"/(1 - "x")`
APPEARS IN
RELATED QUESTIONS
Find the derivative of the function y = f(x) using the derivative of the inverse function x = f–1(y) in the following:
y = `sqrt(x)`
Find the derivative of the function y = f(x) using the derivative of the inverse function x = f-1(y) in the following: y = `sqrt(2 - sqrt(x)`
Find the derivative of the function y = f(x) using the derivative of the inverse function x = f–1(y) in the following: y = 2x + 3
Find the derivative of the function y = f(x) using the derivative of the inverse function x = f–1(y) in the following: y = e2x-3
Find the derivative of the inverse function of the following : y = x log x
Using derivative, prove that: sec–1x + cosec–1x = `pi/(2)` ...[for |x| ≥ 1]
Find the rate of change of demand (x) of a commodity with respect to its price (y) if y = 18x + log(x - 4).
If g is the inverse of f and f'(x) = `1/(1 + x^4)` then g'(x) = ______
Find the derivative of the inverse of function y = 2x3 – 6x and calculate its value at x = −2
Choose the correct alternative:
What is the rate of change of demand (x) of a commodity with respect to its price (y) if y = 10 + x + 25x3.
State whether the following statement is True or False:
If y = 10x + 1, then `("d"y)/("d"x)` = 10x.log10
State whether the following statement is True or False:
If y = 7x + 1, then the rate of change of demand (x) of a commodity with respect to its price (y) is 7
Find the rate of change of demand (x) of a commodity with respect to its price (y) if y = 5 + x2e–x + 2x
I.F. of dx = y (x + y ) dy is a function of ______.
Find `dy/dx`, if y = `sec^-1((1 + x^2)/(1 - x^2))`.
If y = `cos^-1 sqrt((1 + x^2)/2`, then `dy/dx` = ______.
Find the rate of change of demand (x) of a commodity with respect to its price (y) if y = 12 + 10x + 25x2.
Find the rate of change of demand (x) of a commodity with respect to its price (y) if `y=12+10x+25x^2`
