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 = 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 = log10 x
Differentiate the following:
y = tan 3x
Differentiate the following:
y = cos (tan x)
Differentiate the following:
y = (2x – 5)4 (8x2 – 5)–3
Differentiate the following:
y = `x"e"^(-x^2)`
Differentiate the following:
y = `(sin^2x)/cos x`
Differentiate the following:
y = `5^((-1)/x)`
Find the derivatives of the following:
tan (x + y) + tan (x – y) = x
Find the derivatives of the following:
`cos[2tan^-1 sqrt((1 - x)/(1 + x))]`
Find the derivatives of the following:
Find the derivative of sin x2 with respect to x2
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^-1 x)/sqrt(1 - x^2)`, show that (1 – x2)y2 – 3xy1 – y = 0
Choose the correct alternative:
`"d"/("d"x) (2/pi sin x^circ)` 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) = `{{:(2"a" - x, "for" - "a" < x < "a"),(3x - 2"a", "for" x ≥ "a"):}` , then which one of the following is true?
Choose the correct alternative:
If f(x) = `{{:("a"x^2 - "b"",", - 1 < x < 1),(1/|x|",", "elsewhere"):}` is differentiable at x = 1, then
