Advertisements
Advertisements
Question
If `"x"^"a"*"y"^"b" = ("x + y")^("a + b")`, then show that `"dy"/"dx" = "y"/"x"`
Solution
`"x"^"a"*"y"^"b" = ("x + y")^("a + b")`
Taking logarithm of both sides, we get
log (`"x"^"a"*"y"^"b"`) = log `("x + y")^("a + b")`
∴ log `"x"^"a" + log "y"^"b" = ("a + b") log ("x + y")`
∴ a log x + b log y = (a + b) log (x + y)
Differentiating both sides w.r.t. x, we get
`"a"(1/"x") + "b"(1/"y") "dy"/"dx" = ("a + b")(1/("x + y")) "d"/"dx" ("x + y")`
∴ `"a"/"x" + "b"/"y" "dy"/"dx" = ("a + b")/("x + y") (1 + "dy"/"dx")`
∴ `"a"/"x" + "b"/"y" "dy"/"dx" = ("a + b")/("x + y") + ("a + b")/("x + y") "dy"/"dx"`
∴ `"b"/"y" "dy"/"dx" - ("a + b")/("x + y") "dy"/"dx" = ("a + b")/("x + y") - "a"/"x"`
∴ `("b"/"y" - ("a + b")/("x + y")) "dy"/"dx" = ("a + b")/("x + y") - "a"/"x"`
∴ `[("bx" + "by" - "a""y" - "by")/("y"("x + y"))] "dy"/"dx" = ("ax" + "bx" - "ax" - "ay")/("x"("x + y"))`
∴ `[("bx" - "ay")/("y"("x + y"))] "dy"/"dx" = ("bx" - "ay")/("x"("x + y"))`
∴ `"dy"/"dx" = ("bx" - "ay")/("x"("x + y")) xx ("y"("x + y"))/("bx" - "ay")`
∴ `"dy"/"dx" = "y"/"x"`
APPEARS IN
RELATED QUESTIONS
Find `dy/dx` in the following:
2x + 3y = sin x
Write the derivative of f (x) = |x|3 at x = 0.
Find `dy/dx if x^3 + y^2 + xy = 7`
Find `"dy"/"dx"` ; if y = cos-1 `("2x" sqrt (1 - "x"^2))`
Differentiate e4x + 5 w.r..t.e3x
Find `"dy"/"dx"`, if : x = sinθ, y = tanθ
Find `"dy"/"dx"`, if : `x = cos^-1((2t)/(1 + t^2)), y = sec^-1(sqrt(1 + t^2))`
Find `"dy"/"dx"`, if : `x = cos^-1(4t^3 - 3t), y = tan^-1(sqrt(1 - t^2)/t)`.
Differentiate `sin^-1((2x)/(1 + x^2))w.r.t. cos^-1((1 - x^2)/(1 + x^2))`
Differentiate `tan^-1((sqrt(1 + x^2) - 1)/(x)) w.r.t tan^-1((2xsqrt(1 - x^2))/(1 - 2x^2))`.
Find `(d^2y)/(dx^2)` of the following : x = sinθ, y = sin3θ at θ = `pi/(2)`
If y = eax.sin(bx), show that y2 – 2ay1 + (a2 + b2)y = 0.
Find the nth derivative of the following : eax+b
Find the nth derivative of the following : apx+q
Find the nth derivative of the following : cos x
Find the nth derivative of the following : `(1)/(3x - 5)`
Choose the correct option from the given alternatives :
If y = sin (2sin–1 x), then dx = ........
Choose the correct option from the given alternatives :
If y = `tan^-1(x/(1 + sqrt(1 - x^2))) + sin[2tan^-1(sqrt((1 - x)/(1 + x)))] "then" "dy"/"dx"` = ...........
Differentiate the following w.r.t. x : `sin[2tan^-1(sqrt((1 - x)/(1 + x)))]`
Differentiate the following w.r.t. x : `cos^-1((sqrt(1 + x) - sqrt(1 - x))/2)`
Differentiate the following w.r.t. x : `tan^-1[sqrt((sqrt(1 + x^2) + x)/(sqrt(1 + x^2) - x))]`
Differentiate log `[(sqrt(1 + x^2) + x)/(sqrt(1 + x^2 - x)]]` w.r.t. cos (log x).
If log y = log (sin x) – x2, show that `(d^2y)/(dx^2) + 4x "dy"/"dx" + (4x^2 + 3)y` = 0.
Find `"dy"/"dx"` if, xy = log (xy)
Solve the following:
If `"e"^"x" + "e"^"y" = "e"^((x + y))` then show that, `"dy"/"dx" = - "e"^"y - x"`.
If x2 + y2 = 1, then `(d^2x)/(dy^2)` = ______.
If x = a t4 y = 2a t2 then `("d"y)/("d"x)` = ______
`(dy)/(dx)` of `xy + y^2 = tan x + y` is
y = `e^(x3)`
If y = `sqrt(tan x + sqrt(tanx + sqrt(tanx + .... + ∞)`, then show that `dy/dx = (sec^2x)/(2y - 1)`.
Find `dy/dx` at x = 0.
Find `dy/dx if, x= e^(3t), y = e^sqrtt`
`"If" log(x+y) = log(xy)+a "then show that", dy/dx=(-y^2)/x^2`
If log(x + y) = log(xy) + a then show that, `dy/dx = (-y^2)/x^2`
Find `dy/dx` if, x = `e^(3t)`, y = `e^sqrtt`
If y = `(x + sqrt(x^2 - 1))^m`, show that `(x^2 - 1)(d^2y)/(dx^2) + xdy/dx` = m2y
Find `dy/dx` if, `x = e^(3t), y = e^sqrtt`
If log(x + y) = log(xy) + a then show that, `dy/dx = (−y^2)/x^2`
If log(x + y) = log(xy) + a then show that, `dy/dx = (-y^2)/x^2`
