Advertisements
Advertisements
Question
Find the derivatives of the following:
If y = `(cos^-1 x)^2`, prove that `(1 - x^2) ("d"^2y)/("d"x)^2 - x ("d"y)/("d"x) - 2` = 0. Hence find y2 when x = 0
Advertisements
Solution
y = `(cos^-1 x)^2`
`("d"y)/("d"x) = 2(cos^-1_x) xx 1/(- sqrt(1 - x^2)`
`sqrt(1 - x^2) ("d"y)/("d"x) = - 2 cos^-1 x`
`sqrt(1 - x^2) * ("d"^2y)/("d"x)^2 + ("d"y)/("d"x) xx 1/2 (1 - x^2)^(1/2 - 1) (- 2x) = - 2xx 1/sqrt(1 - x^2)`
`sqrt(1 - x^2) ("d"^2y)/("d"x^2) - x (1 - x^2)^(1/2) ("d"y)/("d"x) = 2/sqrt(1 - x^2)`
`sqrt(1 - x^2) ("d"^2y)/("d"x^2) - x/sqrt(1 - x^2) ("d"y)/("d"x) = 2/sqrt(1 - x^2)`
`sqrt(1 - x^2) [sqrt(1 - x^2) ("d"^2y)/("d"x) - x/sqrt(1 - x^2) ("d"y)/("d"x)]` = 2
`(1 - x^2) ("d"^2y)/("d"x^2) - x * ("d"y)/("d"x)` = 2
`(1 - x^2) ("d"^2y)/("d"x^2) - x ("d"y)/("d"x) - 2` = 0
APPEARS IN
RELATED QUESTIONS
Find the derivatives of the following functions with respect to corresponding independent variables:
y = `x/(sin x + cosx)`
Find the derivatives of the following functions with respect to corresponding independent variables:
y = x sin x cos x
Differentiate the following:
y = cos (tan x)
Differentiate the following:
y = `"e"^sqrt(x)`
Differentiate the following:
y = sin (ex)
Differentiate the following:
f(t) = `root(3)(1 + tan "t")`
Differentiate the following:
y = cos (a3 + x3)
Differentiate the following:
y = (2x – 5)4 (8x2 – 5)–3
Differentiate the following:
y = `x"e"^(-x^2)`
Differentiate the following:
f(x) = `x/sqrt(7 - 3x)`
Differentiate the following:
y = `(sin^2x)/cos x`
Differentiate the following:
y = `5^((-1)/x)`
Find the derivatives of the following:
xy = yx
Find the derivatives of the following:
`x^2/"a"^2 + y^2/"b"^2` = 1
Find the derivatives of the following:
`sqrt(x^2 + y^2) = tan^-1 (y/x)`
Find the derivatives of the following:
sin-1 (3x – 4x3)
Choose the correct alternative:
If y = cos (sin x2), then `("d"y)/("d"x)` at x = `sqrt(pi/2)` is
Choose the correct alternative:
If the derivative of (ax – 5)e3x at x = 0 is – 13, then the value of a is
Choose the correct alternative:
If x = a sin θ and y = b cos θ, then `("d"^2y)/("d"x^2)` is
