Advertisements
Advertisements
Question
Find the derivatives of the following:
Find the derivative with `tan^-1 ((sinx)/(1 + cos x))` with respect to `tan^-1 ((cosx)/(1 + sinx))`
Advertisements
Solution
Let u = `tan^-1 ((sinx)/(1 + cos x))`
= `tan^-1 ((2 sin x/2 cos x/2)/(2 cos ^2 x/2))`
= `tan^-1 ((sin x/2)/(cos x/2))`
= `tan^-1 (tan x/2)`
u = `x/2`
`("d"u)/("d"x) = 1/2` .......(1)
v = `tan^-1 [cos x/(1 + sin x)]`
= `tan^-1 [(cos^2 x/2 - sin^2 x/2)/(cos^2 x/2 + sin^2 x/2 + 2 sin x/2 cos x/2)]`
= `tan^-1 [((cos x/2 + sin x/2)(cos x/2 - sin x/2))/(cos x/2 + sin x/2)^2]`
= `tan^-1 [(cos x/2 - sin x/2)/(cos x/2 + sin x/2)]`
= `tan^-1 [(cos x/2 (1 - (sin x/2)/(cos x/2)))/(cos x/2 (1 + (sin x/2)/(cos x/2)))]`
= `tan^-1 [(1 - tan x/2)/(1 + tan x/2)]`
= `tan^-1 [(tan pi/4 - tan pi/2)/(1 + tan pi/4 * tan x/2)]`
= `tan^-1 [tan (pi/4 - x/2)]`
v = `pi/4 - x/2`
`("d"v)/("d"x) = - 1/2` .........(2)
From equations (1) and (2)
`(("d"u)/("d"x))/(("d"v)/("d"x)) = (1/2)/(- 1/2)`
`("d"u)/("d"v)` = – 1
`("d"tan^-1 [(sin x)/(1 + cos x)])/("d"tan^-1 [cos x/(1 + sin x)])` = – 1
APPEARS IN
RELATED QUESTIONS
Find the derivatives of the following functions with respect to corresponding independent variables:
f(x) = x – 3 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 = cosec x . cot x
Find the derivatives of the following functions with respect to corresponding independent variables:
y = e-x . log x
Differentiate the following:
y = `root(3)(1 + x^3)`
Differentiate the following:
F(x) = (x3 + 4x)7
Differentiate the following:
y = tan (cos x)
Differentiate the following:
y = `sqrt(1 + 2tanx)`
Differentiate the following:
y = `sqrt(x + sqrt(x + sqrt(x)`
Differentiate the following:
y = `sin^-1 ((1 - x^2)/(1 + x^2))`
Find the derivatives of the following:
y = `x^(logx) + (logx)^x`
Find the derivatives of the following:
xy = yx
Find the derivatives of the following:
`tan^-1 = ((6x)/(1 - 9x^2))`
Find the derivatives of the following:
x = `(1 - "t"^2)/(1 + "t"^2)`, y = `(2"t")/(1 + "t"^2)`
Find the derivatives of the following:
`tan^-1 ((cos x + sin x)/(cos x - sin x))`
Find the derivatives of the following:
If u = `tan^-1 (sqrt(1 + x^2) - 1)/x` and v = `tan^-1 x`, find `("d"u)/("d"v)`
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
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"`
Choose the correct alternative:
If y = `1/4 u^4`, u = `2/3 x^3 + 5`, then `("d"y)/("d"x)` is
