Advertisements
Advertisements
Question
Find `dy/dx if x^2y^2 - tan^-1(sqrt(x^2 + y^2)) = cot^-1(sqrt(x^2 + y^2))`
Advertisements
Solution
`x^2y^2 - tan^-1(sqrt(x^2 + y^2)) = cot^-1(sqrt(x^2 + y^2))`
∴ `x^2y^2 = tan^-1(sqrt(x^2 + y^2)) + cot^-1(sqrt(x^2 + y^2))`
∴ `x^2y^2 = π/2 ...[∵ tan^-1 x + cot^-1 x = π/2]`
Differentiating both sides w.r.t. x, we get,
`x^2.d/dx(y^2) + y^2.d/dx(x^2) = 0`
∴ `x^2 × 2y dy/dx + y^2 × 2x = 0`
∴ `2x^2y dy/dx = – 2xy^2`
∴ `x dy/dx = – y`
∴ `dy/dx = -y/x`.
APPEARS IN
RELATED QUESTIONS
Find `dy/dx if x + sqrt(xy) + y = 1`
Find `"dy"/"dx"`If x3 + x2y + xy2 + y3 = 81
Find the second order derivatives of the following : e2x . tan x
Find the second order derivatives of the following : xx
Find `"dy"/"dx"` if, y = (5x3 - 4x2 - 8x)9
Find `"dy"/"dx"` if, y = log(log x)
Find `"dy"/"dx"` if, y = `"e"^(5"x"^2 - 2"x" + 4)`
If `"x"^"m"*"y"^"n" = ("x + y")^("m + n")`, then `"dy"/"dx" = "______"/"x"`
`d/dx(10^x) = x*10^(x - 1)`
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 `"dy"/"dx"`, if y = xx.
Differentiate `"e"^("4x" + 5)` with respect to 104x.
If sin−1(x3 + y3) = a then `("d"y)/("d"x)` = ______
If y = cos−1 [sin (4x)], find `("d"y)/("d"x)`
If x = f(t) and y = g(t) are differentiable functions of t so that y is a differentiable function of x and `(dx)/(dt)` ≠ 0 then `(dy)/(dx) = ((dy)/(dt))/((dx)/(d"))`.
Hence find `(dy)/(dx)` if x = sin t and y = cost
Choose the correct alternative:
If y = `x^(sqrt(x))`, then `("d"y)/("d"x)` = ?
If y = x10, then `("d"y)/("d"x)` is ______
If y = `("e")^((2x + 5))`, then `("d"y)/("d"x)` is ______
If y = x2, then `("d"^2y)/("d"x^2)` is ______
Find `("d"^2y)/("d"x^2)`, if y = `"e"^((2x + 1))`
Find `("d"y)/("d"x)`, if y = `root(5)((3x^2 + 8x + 5)^4`
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}` = ______
Derivative of ex sin x w.r.t. e-x cos x is ______.
`"d"/("d"x) [sin(1 - x^2)]^2` = ______.
If y = `(cos x)^((cosx)^((cosx))`, then `("d")/("d"x)` = ______.
Differentiate `sqrt(tansqrt(x))` w.r.t. x
Find `("d"y)/("d"x)`, if y = `tan^-1 ((3x - x^3)/(1 - 3x^2)), -1/sqrt(3) < x < 1/sqrt(3)`
If f(x) = |cos x|, find f'`((3pi)/4)`
If y = `sec^-1 ((sqrt(x) + 1)/(sqrt(x + 1))) + sin^-1((sqrt(x) - 1)/(sqrt(x) + 1))`, then `"dy"/"dx"` is equal to ______.
y = `2sqrt(cotx^2)`
y = `cos sqrt(x)`
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 ______.
If f(x) = `{{:(x^3 + 1",", x < 0),(x^2 + 1",", x ≥ 0):}`, g(x) = `{{:((x - 1)^(1//3)",", x < 1),((x - 1)^(1//2)",", x ≥ 1):}`, then (gof) (x) is equal to ______.
If `d/dx` [f(x)] = ax+ b and f(0) = 0, then f(x) is equal to ______.
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)`
Find `dy/dx` if, `y=e^(5x^2-2x+4)`
If `y=root5((3x^2+8x+5)^4)`, find `dy/dx`
Solve the following:
If y = `root(5)((3"x"^2 + 8"x" + 5)^4)`, find `"dy"/"dx"`
If `y = root{5}{(3x^2 + 8x + 5)^4}, "find" dy/dx`.
If y = `root{5}{(3x^2 + 8x + 5)^4)`, find `(dy)/(dx)`
Find `dy/dx` if, `y = e^(5x^2 - 2x + 4)`.
