Advertisements
Advertisements
प्रश्न
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
उत्तर
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
संबंधित प्रश्न
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 = `sinx/x^2`
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 = sin x0
Differentiate the following:
y = `root(3)(1 + x^3)`
Differentiate the following:
F(x) = (x3 + 4x)7
Differentiate the following:
h(t) = `("t" - 1/"t")^(3/2)`
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:
tan (x + y) + tan (x – y) = x
Find the derivatives of the following:
sin-1 (3x – 4x3)
Find the derivatives of the following:
Find the derivative of sin x2 with respect to x2
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 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 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
Choose the correct alternative:
If f(x) = `{{:(x - 5, "if" x ≤ 1),(4x^2 - 9, "if" 1 < x < 2),(3x + 4, "if" x ≥ 2):}` , then the right hand derivative of f(x) at 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?
