Advertisements
Advertisements
Question
Differentiate the following w.r.t. x : `cot^-1((1 - sqrt(x))/(1 + sqrt(x)))`
Advertisements
Solution
Let y = `cot^-1((1 - sqrt(x))/(1 + sqrt(x)))`
= `tan^-1((1 + sqrt(x))/(1 - sqrt(x))) ...[∵ cot^-1 x = tan^-1(1/x)]`
= `tan^-1((1 + sqrt(x))/(1 - 1 xx sqrt(x)))`
= `tan^-1(1) + tan^-1(sqrt(x)) ...[∵ tan^-1((x + y)/(1 - xy)) = tan^-1x + tan^-1y]`
= `pi/(4) + tan^-1(sqrt(x))`
Differentiating w.r.t. x, we get
`"dy"/"dx" = "d"/"dx"[pi/4 + tan^-1(sqrt(x))]`
= `"d"/"dx"(pi/4) + "d"/"dx"[tan^-1(sqrt(x))]`
= `0 + (1)/(1 + (sqrt(x))^2)."d"/"dx"(sqrt(x))`
= `(1)/(1 + x) xx (1)/(2sqrt(x)`
= `(1)/(2sqrt(x)(1 + x)`.
APPEARS IN
RELATED QUESTIONS
Differentiate the following w.r.t. x: `sqrt(x^2 + 4x - 7)`.
Differentiate the following w.r.t.x: `log[tan(x/2)]`
Differentiate the following w.r.t.x: `sqrt(tansqrt(x)`
Differentiate the following w.r.t.x: log[cos(x3 – 5)]
Differentiate the following w.r.t.x: cos2[log(x2 + 7)]
Differentiate the following w.r.t.x:
tan[cos(sinx)]
Differentiate the following w.r.t.x: `e^(log[(logx)^2 - logx^2]`
Differentiate the following w.r.t.x: `sinsqrt(sinsqrt(x)`
Differentiate the following w.r.t.x:
sin2x2 – cos2x2
Differentiate the following w.r.t.x:
`log(sqrt((1 - cos3x)/(1 + cos3x)))`
Differentiate the following w.r.t. x : cot–1(4x)
Differentiate the following w.r.t. x:
`sin^-1(sqrt((1 + x^2)/2))`
Differentiate the following w.r.t. x : `sin^4[sin^-1(sqrt(x))]`
Differentiate the following w.r.t. x : `"cosec"^-1[1/cos(5^x)]`
Differentiate the following w.r.t. x : `tan^-1((cos7x)/(1 + sin7x))`
Differentiate the following w.r.t. x : `cos^-1((sqrt(3)cosx - sinx)/(2))`
Differentiate the following w.r.t. x :
`cos^-1[(3cos(e^x) + 2sin(e^x))/sqrt(13)]`
Differentiate the following w.r.t. x:
`sin^-1 ((1 - 25x^2)/(1 + 25x^2))`
Differentiate the following w.r.t. x :
`sin^(−1) ((1 − x^3)/(1 + x^3))`
Differentiate the following w.r.t. x : `tan^-1((2^x)/(1 + 2^(2x + 1)))`
Differentiate the following w.r.t. x : `tan^-1((a + btanx)/(b - atanx))`
Differentiate the following w.r.t. x : `cot^-1((a^2 - 6x^2)/(5ax))`
Differentiate the following w.r.t. x :
`(x + 1)^2/((x + 2)^3(x + 3)^4`
Differentiate the following w.r.t. x : `root(3)((4x - 1)/((2x + 3)(5 - 2x)^2)`
Differentiate the following w.r.t. x : `(x^5.tan^3 4x)/(sin^2 3x)`
Differentiate the following w.r.t. x : (sin x)x
Differentiate the following w.r.t. x: (sin xx)
Differentiate the following w.r.t. x :
etanx + (logx)tanx
Show that `"dy"/"dx" = y/x` in the following, where a and p are constants : `sec((x^5 + y^5)/(x^5 - y^5))` = a2
Show that `"dy"/"dx" = y/x` in the following, where a and p are constants : `cos^-1((7x^4 + 5y^4)/(7x^4 - 5y^4)) = tan^-1a`
Show that `"dy"/"dx" = y/x` in the following, where a and p are constants: `log((x^20 - y^20)/(x^20 + y^20))` = 20
Show that `"dy"/"dx" = y/x` in the following, where a and p are constants : `e^((x^7 - y^7)/(x^7 + y^7)` = a
If y = sin−1 (2x), find `("d"y)/(""d"x)`
If y = `sin^-1[("a"cosx - "b"sinx)/sqrt("a"^2 + "b"^2)]`, then find `("d"y)/("d"x)`
If `t = v^2/3`, then `(-v/2 (df)/dt)` is equal to, (where f is acceleration) ______
If y = `1 + x + x^2/(2!) + x^3/(3!) + x^4/(4!) + .....,` then `(d^2y)/(dx^2)` = ______
If y = `(3x^2 - 4x + 7.5)^4, "then" dy/dx` is ______
If f(x) = `(3x + 1)/(5x - 4)` and t = `(5 + 3x)/(x - 4)`, then f(t) is ______
Let f(x) = `(1 - tan x)/(4x - pi), x ne pi/4, x ∈ [0, pi/2]`. If f(x) is continuous in `[0, pi/2]`, then f`(pi/4)` is ______.
Solve `x + y (dy)/(dx) = sec(x^2 + y^2)`
The value of `d/(dx)[tan^-1((a - x)/(1 + ax))]` is ______.
If `cos((x^2 - y^2)/(x^2 + y^2))` = log a, show that `dy/dx = y/x`
Diffierentiate: `tan^-1((a + b cos x)/(b - a cos x))` w.r.t.x.
`lim_(x → 0) (sqrt(1 + x + x^2) − 1)/x` = ______.
