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
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:
f(x) = x sin x
Find the derivatives of the following functions with respect to corresponding independent variables:
y = cos x – 2 tan x
Find the derivatives of the following functions with respect to corresponding independent variables:
g(t) = t3 cos t
Find the derivatives of the following functions with respect to corresponding independent variables:
y = `sinx/x^2`
Find the derivatives of the following functions with respect to corresponding independent variables:
y = tan θ (sin θ + cos θ)
Find the derivatives of the following functions with respect to corresponding independent variables:
y = cosec x . cot x
Find the derivatives of the following functions with respect to corresponding independent variables:
y = x sin x cos x
Find the derivatives of the following functions with respect to corresponding independent variables:
y = e-x . log x
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:
y = `root(3)(1 + x^3)`
Differentiate the following:
h(t) = `("t" - 1/"t")^(3/2)`
Differentiate the following:
s(t) = `root(4)(("t"^3 + 1)/("t"^3 - 1)`
Differentiate the following:
f(x) = `x/sqrt(7 - 3x)`
Differentiate the following:
y = `"e"^(3x)/(1 + "e"^x`
Find the derivatives of the following:
(cos x)log x
Find the derivatives of the following:
If y = sin–1x then find y”
Find the derivatives of the following:
If y = `(sin^-1 x)/sqrt(1 - x^2)`, show that (1 – x2)y2 – 3xy1 – y = 0
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?
