Advertisements
Advertisements
Question
Find `"dy"/"dx"`If x3 + x2y + xy2 + y3 = 81
Advertisements
Solution
x3 + x2y + xy2 + y3 = 81
Differentiating both sides w.r.t. x, we get
`3x^2 + x^2 "dy"/"dx" + y"d"/"dx"(x^2) + x"d"/"dx"(y^2) + y^2"d"/"dx"(x) + 3y^2"dy"/"dx"` = 0
∴ `3x^2 + x^2"dy"/"dx" + y xx 2x + x xx 2y"dy"/"dx" + y^2 xx 1 + 3y^2"dy"/"dx"` = 0
∴ `3x^2 + x^2"dy"/"dx" + 2xy + 2xy"dy"/"dx" + y^2 + 3y^2"dy"/"dx"` = 0
∴ `(x^2 + 2xy + 3y^2)"dy"/"dx" = -3x^2 - 2xy - y^2`
∴ `"dy"/"dx" = (-(3x^2 + 2xy + y^2))/(x^2 + 2xy + 3y^2)`.
APPEARS IN
RELATED QUESTIONS
If y = eax. cos bx, then prove that
`(d^2y)/(dx^2) - 2ady/dx + (a^2 + b^2)y` = 0
if `y = tan^2(log x^3)`, find `(dy)/(dx)`
Solve the following differential equation:
x2 dy + (xy + y2) dx = 0, when x = 1 and y = 1
Find `dy/dx`, if `sqrt(x) + sqrt(y) = sqrt(a)`.
Find the second order derivatives of the following : e4x. cos 5x
Find `"dy"/"dx"` if, y = (5x3 - 4x2 - 8x)9
Find `"dy"/"dx"` if, y = `"e"^(5"x"^2 - 2"x" + 4)`
Choose the correct alternative.
If y = `sqrt("x" + 1/"x")`, then `"dy"/"dx" = ?`
Solve the following:
If y = (6x3 - 3x2 - 9x)10, find `"dy"/"dx"`
If y = `root(5)((3"x"^2 + 8"x" + 5)^4)`, find `"dy"/"dx"`.
Find the rate of change of demand (x) of a commodity with respect to its price (y) if y = 25 + 30x – x2.
Differentiate `"e"^("4x" + 5)` with respect to 104x.
If f'(4) = 5, f(4) = 3, g'(6) = 7 and R(x) = g[3 + f(x)] then R'(4) = ______
If y = cos−1 [sin (4x)], find `("d"y)/("d"x)`
Choose the correct alternative:
If y = `root(3)((3x^2 + 8x - 6)^5`, then `("d"y)/("d"x)` = ?
State whether the following statement is True or False:
If y = ex, then `("d"^2y)/("d"x^2)` = ex
Find `("d"^2y)/("d"x^2)`, if y = `"e"^((2x + 1))`
If y = `2/(sqrt(a^2 - b^2))tan^-1[sqrt((a - b)/(a + b)) tan x/2], "then" (d^2y)/dx^2|_{x = pi/2}` = ______
If u = x2 + y2 and x = s + 3t, y = 2s - t, then `(d^2u)/(ds^2)` = ______
If y = `x/"e"^(1 + x)`, then `("d"y)/("d"x)` = ______.
If y = (sin x2)2, then `("d"y)/("d"x)` is equal to ______.
If y = `(cos x)^((cosx)^((cosx))`, then `("d")/("d"x)` = ______.
If ex + ey = ex+y , prove that `("d"y)/("d"x) = -"e"^(y - x)`
If f(x) = |cos x – sinx|, find `"f'"(pi/6)`
If y = `sec^-1 ((sqrt(x) + 1)/(sqrt(x + 1))) + sin^-1((sqrt(x) - 1)/(sqrt(x) + 1))`, then `"dy"/"dx"` is equal to ______.
Let f(x) = log x + x3 and let g(x) be the inverse of f(x), then |64g"(1)| is equal to ______.
Let x(t) = `2sqrt(2) cost sqrt(sin2t)` and y(t) = `2sqrt(2) sint sqrt(sin2t), t ∈ (0, π/2)`. Then `(1 + (dy/dx)^2)/((d^2y)/(dx^2)` at t = `π/4` is equal to ______.
Solve the following:
If y = `root5 ((3x^2 + 8x + 5)^4 ,) "find" "dy"/ "dx"`
Find `"dy"/"dx"` if, `"y" = "e"^(5"x"^2 - 2"x" + 4)`
Find `dy/dx` if, y = `e^(5x^2 - 2x + 4)`
Solve the following:
If y = `root5((3x^2 + 8x + 5)^4)`, find `dy/dx`
If y = `root5((3x^2 + 8x + 5)^4)`, find `dy/dx`
Find `dy/dx` if ,
`x= e^(3t) , y = e^(4t+5)`
If y = `log((x + sqrt(x^2 + a^2))/(sqrt(x^2 + a^2) - x))`, find `dy/dx`.
Solve the following:
If y = `root5((3x^2 + 8x + 5)^4)`, find `dy/dx`
If `y=root5((3x^2+8x+5)^4)`, find `dy/dx`
Find `"dy"/"dx"` if, y = `"e"^(5"x"^2 - 2"x" + 4)`
Solve the following:
If y = `root(5)((3"x"^2 + 8"x" + 5)^4)`, find `"dy"/"dx"`
Find `dy/dx` if, `y = e^(5x^2 - 2x + 4)`.
Solve the following:
If y = `root5((3x^2 + 8x + 5)^4)`, find `dy/(dx)`.
