Advertisements
Advertisements
प्रश्न
Find the derivatives of the following:
x = `(1 - "t"^2)/(1 + "t"^2)`, y = `(2"t")/(1 + "t"^2)`
Advertisements
उत्तर
x = `(1 - "t"^2)/(1 + "t"^2)`, y = `(2"t")/(1 + "t"^2)`
Put t = tan θ
x = `(1 - tan^2theta)/(1 + tan^2theta)` y = `(2tantheta)/(1 + tan^2theta)`
x = cos 2θ, y = sin 2θ
`("d"x)/("d"theta) = - sin 2theta xx 2`, `("d"y)/("d"theta) = cos 2theta xx 2`
`("d"x)/("d"theta) = - 2 sin 2theta`, `("d"y)/("d"theta) = 2 cos 2theta`
`(("d"y)/("d"theta))/(("d"x)/("d"theta)) = (2 cos 2theta)/(-2 sin 2theta)`
`("d"y)/("d"x) = - cot 2theta`
tan 2θ = `(2tantheta)/(1 - tan^2theta)`
cot 2θ = `(1 - tan^2theta)/(2tantheta)`
= `- (1 - tan^2theta)/(2tantheta)`
= `- (1 - "t"^2)/(2"t")`
`("d"y)/("d"x) = ("t"^2 - 1)/(2"t")`
APPEARS IN
संबंधित प्रश्न
Find the derivatives of the following functions with respect to corresponding independent variables:
y = `tan x/x`
Differentiate the following:
y = cos (tan x)
Differentiate the following:
f(t) = `root(3)(1 + tan "t")`
Differentiate the following:
y = `(sin^2x)/cos x`
Differentiate the following:
y = `5^((-1)/x)`
Differentiate the following:
y = sin2(cos kx)
Differentiate the following:
y = (1 + cos2)6
Differentiate the following:
y = `"e"^(3x)/(1 + "e"^x`
Differentiate the following:
y = `"e"^(xcosx)`
Find the derivatives of the following:
y = `x^(logx) + (logx)^x`
Find the derivatives of the following:
tan (x + y) + tan (x – y) = x
Find the derivatives of the following:
If cos(xy) = x, show that `(-(1 + ysin(xy)))/(xsiny)`
Find the derivatives of the following:
`tan^-1 = ((6x)/(1 - 9x^2))`
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 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
