Advertisements
Advertisements
Question
If `sec^-1((7x^3 - 5y^3)/(7^3 + 5y^3)) = "m", "show" (d^2y)/(dx^2)` = 0.
Advertisements
Solution
`sec^-1((7x^3 - 5y^3)/(7^3 + 5y^3))` = m
∴ `(7x^3 - 5y^3)/(7x^3 + 5y^3)` = sec m =k ...(Say)
∴ 7x3 – 5y3 = 7kx3 + 5ky3
∴ (5k + 5)y3 = (7 – 7k)x2
∴ `y^3/x^3 = (7 - 7k)/(5k + 5)`
∴ `y/x = ((7 - 7k)/(5k + 5))^(1/3)` = p, where p is a constant
∴ `"d"/"dx"(y/x) = "d"/"dx"(p)`
∴ `(x"dy"/"dx" - y "d"/"dx"(x))/(x^2)` = 0
∴ `x"dy"/"dx" - y xx 1` = 0
∴ `x"dy"/"dx"` = y
∴ `"dy"/"dx" = y/x` ...(1)
∴ `(d^2y)/(dx^2) = "d"/"dx"(y/x)`
= `(x"dy"/"dx" - y "d"/"dx"(x))/(x^2)`
= `(x(y/x) - y xx 1)/(x^2)` ...[By (1)]
= `(y - y)/x^2`
= `0/x^2`
= 0
Note : `"dy"/"dx" = y/x. "where" y/x` = p.
∴ `"dy"/"dx"` = p, where p is a constant.
∴ `(d^2y)/(dx^2) = "d"/"dx"(p)` = 0.
APPEARS IN
RELATED QUESTIONS
Find `bb(dy/dx)` in the following:
x2 + xy + y2 = 100
if `x^y + y^x = a^b`then Find `dy/dx`
if `(x^2 + y^2)^2 = xy` find `(dy)/(dx)`
If for the function
\[\Phi \left( x \right) = \lambda x^2 + 7x - 4, \Phi'\left( 5 \right) = 97, \text { find } \lambda .\]
Is |sin x| differentiable? What about cos |x|?
Write the derivative of f (x) = |x|3 at x = 0.
Find `dy/dx if x^3 + y^2 + xy = 7`
Find `"dy"/"dx"` ; if x = sin3θ , y = cos3θ
Find `(dy)/(dx) if y = cos^-1 (√x)`
Differentiate tan-1 (cot 2x) w.r.t.x.
Discuss extreme values of the function f(x) = x.logx
If `sin^-1((x^5 - y^5)/(x^5 + y^5)) = pi/(6), "show that" "dy"/"dx" = x^4/(3y^4)`
Find `"dy"/"dx"` if x = a cot θ, y = b cosec θ
Find `"dy"/"dx"` if : x = a cos3θ, y = a sin3θ at θ = `pi/(3)`
Find `"dy"/"dx"` if : x = t + 2sin (πt), y = 3t – cos (πt) at t = `(1)/(2)`
Differentiate `tan^-1((x)/(sqrt(1 - x^2))) w.r.t. sec^-1((1)/(2x^2 - 1))`.
If x = at2 and y = 2at, then show that `xy(d^2y)/(dx^2) + a` = 0.
If y = sin (m cos–1x), then show that `(1 - x^2)(d^2y)/(dx^2) - x"dy"/"dx" + m^2y` = 0.
Find the nth derivative of the following : (ax + b)m
Find the nth derivative of the following : eax+b
Find the nth derivative of the following : cos (3 – 2x)
Find the nth derivative of the following : y = eax . cos (bx + c)
Find the nth derivative of the following:
y = e8x . cos (6x + 7)
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 y = sec (tan –1x), then `"dy"/"dx"` at x = 1, is equal to
Differentiate the following w.r.t. x : `cos^-1((sqrt(1 + x) - sqrt(1 - x))/2)`
Differentiate `tan^-1((sqrt(1 + x^2) - 1)/x)` w.r.t. `cos^-1(sqrt((1 + sqrt(1 + x^2))/(2sqrt(1 + x^2))))`
If log y = log (sin x) – x2, show that `(d^2y)/(dx^2) + 4x "dy"/"dx" + (4x^2 + 3)y` = 0.
If x= a cos θ, y = b sin θ, show that `a^2[y(d^2y)/(dx^2) + (dy/dx)^2] + b^2` = 0.
If y = Aemx + Benx, show that y2 – (m + n)y1 + mny = 0.
Find `"dy"/"dx" if, sqrt"x" + sqrt"y" = sqrt"a"`
Find `"dy"/"dx"` if, x3 + x2y + xy2 + y3 = 81
Find `"dy"/"dx"` if, `"x"^"y" = "e"^("x - y")`
If `"x"^5 * "y"^7 = ("x + y")^12` then show that, `"dy"/"dx" = "y"/"x"`
Choose the correct alternative.
If y = 5x . x5, then `"dy"/"dx" = ?`
If y = `("x" + sqrt("x"^2 - 1))^"m"`, then `("x"^2 - 1) "dy"/"dx"` = ______.
Differentiate w.r.t x (over no. 24 and 25) `e^x/sin x`
y = `e^(x3)`
`"If" log(x+y) = log(xy)+a "then show that", dy/dx=(-y^2)/x^2`
Find `dy/dx` if , x = `e^(3t), y = e^(sqrtt)`
If log (x+y) = log (xy) + a then show that, `dy/dx= (-y^2)/(x^2)`
