Advertisements
Advertisements
प्रश्न
Choose the correct alternative.
If `"x"^4."y"^5 = ("x + y")^("m + 1")` then `"dy"/"dx" = "y"/"x"` then m = ?
विकल्प
8
4
5
20
Advertisements
उत्तर
8
Explanation:
x4. y5 = (x + y)m + 1 ...(i)
∴ `"d"/"dx" ("x"^4. "y"^5) = "d"/"dx" ("x" + "y")^("m" + 1)`
∴ `"x"^4 "d"/"dx" "y"^5 + "y"^5 "d"/"dx" "x"^4 = ("m" + 1)("x" + "y")^("m" + 1 − 1) . "d"/"dx" ("x" + "y")`
∴ `"x"^4 . 5"y"^4 "d"/"dx" "y"+ "y"^5 4"x"^3 "d"/"dx" "x" = ("m" + 1)("x" + "y")^"m" ["d"/"dx" "x" + "d"/"dx" "y"]`
∴ `5"x"^4"y"^4 "dy"/"dx" + 4"x"^3 "y"^5 . 1 = ("m" + 1)("x" + "y")^"m" [1 + "dy"/"dx"]`
∴ `5"x"^4"y"^4 "dy"/"dx" + 4"x"^3 "y"^5 = ("m" + 1)("x" + "y")^"m" [1 + "dy"/"dx"]`
Put `"dy"/"dx" = "y"/"x"`
∴ `5"x"^((cancel4)3)"y"^4 . "y"/cancel"x" + 4"x"^3 "y"^5 = ("m" + 1)("x" + "y")^"m" [1 + "y"/"x"]`
∴ `5"x"^3"y"^4 . "y" + 4"x"^3 "y"^5 = ("m" + 1)("x" + "y")^"m" [("x" + "y")/"x"]`
∴ `5"x"^3"y"^5 + 4"x"^3 "y"^5 = ("m" + 1)("x" + "y")^"m" [("x" + "y")/"x"]`
∴ `9"x"^3"y"^5 = ("m" + 1)/"x" [("x" + "y")^("m" + 1)]`
∴ `9"x"^3"y"^5 = ("m" + 1)/cancel"x" "x"^((cancel4)3)."y"^5`
∴ `9cancel("x"^3"y"^5) = ("m" + 1) cancel("x"^3"y"^5)`
∴ 9 = m + 1
∴ m = 9 - 1
∴ m = 8
APPEARS IN
संबंधित प्रश्न
Find dy/dx if x sin y + y sin x = 0.
Find `bb(dy/dx)` in the following:
ax + by2 = cos y
Find `bb(dy/dx)` in the following:
sin2 x + cos2 y = 1
If ex + ey = ex+y, then show that `"dy"/"dx" = -e^(y - x)`.
Find `"dy"/"dx"`, if : x = `sqrt(a^2 + m^2), y = log(a^2 + m^2)`
Find `"dy"/"dx"`, if : x = sinθ, y = tanθ
Find `"dy"/"dx"` if : x = a cos3θ, y = a sin3θ at θ = `pi/(3)`
Differentiate `cos^-1((1 - x^2)/(1 + x^2)) w.r.t. tan^-1 x.`
If y = sin (m cos–1x), then show that `(1 - x^2)(d^2y)/(dx^2) - x"dy"/"dx" + m^2y` = 0.
If x = a sin t – b cos t, y = a cos t + b sin t, show that `(d^2y)/(dx^2) = -(x^2 + y^2)/(y^3)`.
Find the nth derivative of the following : eax+b
Find the nth derivative of the following : cos x
Find the nth derivative of the following : sin (ax + b)
Choose the correct option from the given alternatives :
Let `f(1) = 3, f'(1) = -(1)/(3), g(1) = -4 and g'(1) =-(8)/(3).` The derivative of `sqrt([f(x)]^2 + [g(x)]^2` w.r.t. x at x = 1 is
Choose the correct option from the given alternatives :
If `xsqrt(y + 1) + ysqrt(x + 1) = 0 and x ≠ y, "then" "dy"/"dx"` = ........
If y `tan^-1(sqrt((a - x)/(a + x)))`, where – a < x < a, then `"dy"/"dx"` = .........
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(x/(1 + 6x^2)) + cot^-1((1 - 10x^2)/(7x))`
If `xsqrt(1 - y^2) + ysqrt(1 - x^2)` = 1, then show that `"dy"/"dx" = -sqrt((1 - y^2)/(1 - x^2)`.
If x = `e^(x/y)`, then show that `dy/dx = (x - y)/(xlogx)`
Find `"dy"/"dx"` if, xy = log (xy)
Choose the correct alternative.
If x = `("e"^"t" + "e"^-"t")/2, "y" = ("e"^"t" - "e"^-"t")/2` then `"dy"/"dx"` = ?
If x = a t4 y = 2a t2 then `("d"y)/("d"x)` = ______
If `sqrt(x) + sqrt(y) = sqrt("a")`, then `("d"y)/("d"x)` is ______
y = `e^(x3)`
`"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 = e3t, y = `e^sqrtt`
If log(x + y) = log(xy) + a then show that, `dy/dx=(-y^2)/x^2`
