Advertisements
Advertisements
प्रश्न
If ax2 + 2hxy + by2 = 0, then show that `("d"^2"y")/"dx"^2` = 0
Advertisements
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
Find the second order derivative of the function.
x3 log x
Find the second order derivative of the function.
e6x cos 3x
If y = 3 cos (log x) + 4 sin (log x), show that x2y2 + xy1 + y = 0.
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`.
Find `("d"^2"y")/"dx"^2`, if y = `sqrt"x"`
Find `("d"^2"y")/"dx"^2`, if y = `"x"^-7`
Find `("d"^2"y")/"dx"^2`, if y = `"e"^((2"x" + 1))`.
If x2 + 6xy + y2 = 10, then show that `("d"^2y)/("d"x^2) = 80/(3x + y)^3`
`sin xy + x/y` = x2 – y
If ax2 + 2hxy + by2 + 2gx + 2fy + c = 0, then show that `"dy"/"dx" * "dx"/"dy"` = 1
If x sin (a + y) + sin a cos (a + y) = 0, prove that `"dy"/"dx" = (sin^2("a" + y))/sin"a"`
If y = tan–1x, find `("d"^2y)/("dx"^2)` in terms of y alone.
The derivative of cos–1(2x2 – 1) w.r.t. cos–1x is ______.
If x2 + y2 + sin y = 4, then the value of `(d^2y)/(dx^2)` at the point (–2, 0) is ______.
Let for i = 1, 2, 3, pi(x) be a polynomial of degree 2 in x, p'i(x) and p''i(x) be the first and second order derivatives of pi(x) respectively. Let,
A(x) = `[(p_1(x), p_1^'(x), p_1^('')(x)),(p_2(x), p_2^'(x), p_2^('')(x)),(p_3(x), p_3^'(x), p_3^('')(x))]`
and B(x) = [A(x)]T A(x). Then determinant of B(x) ______
If y = `sqrt(ax + b)`, prove that `y((d^2y)/dx^2) + (dy/dx)^2` = 0.
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))`
