Advertisements
Advertisements
प्रश्न
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
उत्तर
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
संबंधित प्रश्न
Find the derivatives of the following functions with respect to corresponding independent variables:
y = `sinx/(1 + cosx)`
Find the derivatives of the following functions with respect to corresponding independent variables:
y = (x2 + 5) log(1 + x) e–3x
Differentiate the following:
y = tan 3x
Differentiate the following:
F(x) = (x3 + 4x)7
Differentiate the following:
h(t) = `("t" - 1/"t")^(3/2)`
Differentiate the following:
y = `x"e"^(-x^2)`
Differentiate the following:
f(x) = `x/sqrt(7 - 3x)`
Differentiate the following:
y = (1 + cos2)6
Find the derivatives of the following:
`tan^-1 = ((6x)/(1 - 9x^2))`
Find the derivatives of the following:
`cos^-1 ((1 - x^2)/(1 + x^2))`
Find the derivatives of the following:
`tan^-1 ((cos x + sin x)/(cos x - sin x))`
Find the derivatives of the following:
If x = a(θ + sin θ), y = a(1 – cos θ) then prove that at θ = `pi/2`, yn = `1/"a"`
Find the derivatives of the following:
If sin y = x sin(a + y), the prove that `("d"y)/("d"x) = (sin^2("a" + y))/sin"a"`, a ≠ nπ
Choose the correct alternative:
If y = `1/("a" - z)`, then `("d"z)/("d"y)` is
Choose the correct alternative:
If y = cos (sin x2), then `("d"y)/("d"x)` at x = `sqrt(pi/2)` is
Choose the correct alternative:
`"d"/("d"x) ("e"^(x + 5log x))` is
Choose the correct alternative:
If x = a sin θ and y = b cos θ, then `("d"^2y)/("d"x^2)` is
Choose the correct alternative:
If f(x) = `{{:(2"a" - x, "for" - "a" < x < "a"),(3x - 2"a", "for" x ≥ "a"):}` , then which one of the following is true?
Choose the correct alternative:
If f(x) = `{{:("a"x^2 - "b"",", - 1 < x < 1),(1/|x|",", "elsewhere"):}` is differentiable at x = 1, then
