Advertisements
Advertisements
प्रश्न
If ax2 + 2hxy + by2 = 0, then show that `("d"^2"y")/"dx"^2` = 0
उत्तर
ax2 + 2hxy + by2 = 0 ....(i)
Differentiating both sides w.r.t. x, we get
`"a"("2x") + "2h" * "d"/"dx" ("xy") + "b"("2y") "dy"/"dx" = 0`
∴ `2"ax" + 2"h" ["x" * "dy"/"dx" + "y"(1)] + 2"by" "dy"/"dx" = 0`
∴ `2"ax" + 2"hx" "dy"/"dx" + 2"hy" + 2"by" "dy"/"dx" = 0`
∴ `2 "dy"/"dx" ("hx" + "by") = - 2"ax" - 2"hy"`
∴ `2 "dy"/"dx" = (-2("ax" + "hy"))/("hx" + "by")`
∴ `"dy"/"dx" = (- ("ax" + "hy"))/("hx" + "by")` ....(i)
ax2 + 2hxy + by2 = 0
∴ ax2 + hxy + hxy + by2 = 0
∴ x(ax + hy) + y(hx + by) = 0
∴ y(hx + by) = - x(ax + hy)
∴ `"y"/"x" = (- ("ax" + "hy"))/("hx" + "by")` ...(ii)
From (i) and (ii), we get
`"dy"/"dx" = "y"/"x"` ....(iii)
Again, differentiating both sides w.r.t. x, we get
`("d"^2"y")/"dx"^2 = ("x" * "dy"/"dx" - "y" * "d"/"dx" ("x"))/"x"^2`
`= ("x" * ("y"/"x") - "y"(1))/"x"^2` ....[From (iii)]
`= ("y - y")/"x"^2`
`= 0/"x"^2`
∴ `("d"^2"y")/"dx"^2 = 0`
APPEARS IN
संबंधित प्रश्न
If x = a sin t and `y = a (cost+logtan(t/2))` ,find `((d^2y)/(dx^2))`
If x = a cos θ + b sin θ, y = a sin θ − b cos θ, show that `y^2 (d^2y)/(dx^2)-xdy/dx+y=0`
Find the second order derivative of the function.
x2 + 3x + 2
Find the second order derivative of the function.
`x^20`
Find the second order derivative of the function.
log x
Find the second order derivative of the function.
x3 log x
Find the second order derivative of the function.
tan–1 x
Find the second order derivative of the function.
sin (log x)
If y = cos–1 x, Find `(d^2y)/dx^2` in terms of y alone.
If y = 500e7x + 600e–7x, show that `(d^2y)/(dx^2) = 49y`
If ey (x + 1) = 1, show that `(d^2y)/(dx^2) =((dy)/(dx))^2`
If y = (tan–1 x)2, show that (x2 + 1)2 y2 + 2x (x2 + 1) y1 = 2
If x7 . y9 = (x + y)16 then show that `"dy"/"dx" = "y"/"x"`
If `x^3y^5 = (x + y)^8` , then show that `(dy)/(dx) = y/x`
Find `("d"^2"y")/"dx"^2`, if y = `"x"^-7`
Find `("d"^2"y")/"dx"^2`, if y = `"x"^2 * "e"^"x"`
`sin xy + x/y` = x2 – y
sec(x + y) = xy
The derivative of cos–1(2x2 – 1) w.r.t. cos–1x is ______.
Derivative of cot x° with respect to x is ____________.
If y = `sqrt(ax + b)`, prove that `y((d^2y)/dx^2) + (dy/dx)^2` = 0.
If y = tan x + sec x then prove that `(d^2y)/(dx^2) = cosx/(1 - sinx)^2`.
Read the following passage and answer the questions given below:
|
The relation between the height of the plant ('y' in cm) with respect to its exposure to the sunlight is governed by the following equation y = `4x - 1/2 x^2`, where 'x' is the number of days exposed to the sunlight, for x ≤ 3.
|
- Find the rate of growth of the plant with respect to the number of days exposed to the sunlight.
- Does the rate of growth of the plant increase or decrease in the first three days? What will be the height of the plant after 2 days?
`"Find" (d^2y)/(dx^2) "if" y=e^((2x+1))`
Find `(d^2y)/dx^2 if, y = e^((2x + 1))`
Find `(d^2y)/dx^2` if, `y = e^((2x + 1))`
Find `(d^2y)/dx^2` if, `y = e^((2x + 1))`
Find `(d^2y)/(dx^2)` if, y = `e^((2x+1))`
Find `(d^2y)/dx^2` if, y = `e^((2x + 1))`
Find `(d^2y)/dx^2` if, `y = e^((2x + 1))`
Find `(d^2y)/dx^2` if, `y = e^((2x+1))`
