Advertisements
Advertisements
Question
Differentiate the following:
y = `sin^-1 ((1 - x^2)/(1 + x^2))`
Advertisements
Solution
y = `sin^-1 ((1 - x^2)/(1 + x^2))`
y = f(g(x)
`("d"y)/("d"x)` = f'(g(x)) . g'(x)
`("d"y)/("d"x) = 1/sqrt(1 - ((1 - x^2)/(1 + x^2))^2) xx "d"/("d"x) ((1 - x^2)/(1 + x^2))`
= `1/sqrt(((1 + x^2)^2 - (1 - x^2)^2)/(1 + x^2)^2) xx ((1 + x^2)(- 2x) - (1 - x^2)(2x))/(1 + x^2)^2`
= `1/sqrt((1 + 2x^2 + x^4 - (1 - 2x^2 + x^4))/((1 + x^2))) xx (-2x - 2x^3 - 2x + 2x^3)/(1 + x^2)^2`
= `(1 + x^2)/sqrt(1 + 2x^2 + x^4 - 1 + 2x^2 - x^4) xx (-4x)/(1 + x^2)^2`
= `(-4x)/(sqrt(4x^2) (1 + x^2))`
`("d"y)/("d"x) = (-4x)/(2x(1 + x^2))`
= `- 2/(1 + x^2)`
APPEARS IN
RELATED QUESTIONS
Find the derivatives of the following functions with respect to corresponding independent variables:
y = sin x + cos x
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:
g(t) = 4 sec t + tan t
Find the derivatives of the following functions with respect to corresponding independent variables:
y = ex sin x
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 = sin x0
Differentiate the following:
y = cos (a3 + x3)
Differentiate the following:
y = e–mx
Differentiate the following:
y = 4 sec 5x
Differentiate the following:
y = `(sin^2x)/cos x`
Differentiate the following:
y = `sqrt(x + sqrt(x + sqrt(x)`
Find the derivatives of the following:
xy = yx
Find the derivatives of the following:
`cos^-1 ((1 - x^2)/(1 + x^2))`
Find the derivatives of the following:
Find the derivative of `sin^-1 ((2x)/(1 + x^2))` with respect to `tan^-1 x`
Find the derivatives of the following:
If y = sin–1x then find y”
Find the derivatives of the following:
If y = etan–1x, show that (1 + x2)y” + (2x – 1)y’ = 0
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 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
Choose the correct alternative:
If y = `1/4 u^4`, u = `2/3 x^3 + 5`, then `("d"y)/("d"x)` is
