Advertisements
Advertisements
Question
Differentiate the following w.r.t. x : `sin^-1 ((1 - 25x^2)/(1 + 25x^2))`
Advertisements
Solution
Let y = `sin^-1 ((1 - 25x^2)/(1 + 25x^2))`
= `sin^-1[(1 - (5x)^2)/(1 + (5x)^2)]`
Put 5x = tanθ.
Then θ = tan–1(5x)
∴ y = `sin^-1((1 - tan^2θ)/(1 + tan^2θ))`
= sin–1(cos2θ)
= `sin^-1[sin(pi/2 - 2θ)]`
= `pi/(2) - 2θ`
= `pi/(2) - 2tan^-1(5x)`
Differentiating w.r.t. x, we get
∴ `"dy"/"dx" = "d"/"dx"[pi/2 - 2tan^-1 (5x)]`
= `"d"/"dx"(pi/2) - 2"d"/"dx"[tan^-1(5x)]`
= `0 - 2 xx (1)/(1 + (5)^2)."d"/"dx"(5x)`
= `(-2)/(1 + 25x^2) xx 5`
= `(-10)/(1 + 25x^2)`.
APPEARS IN
RELATED QUESTIONS
Differentiate the following w.r.t. x: `sqrt(x^2 + 4x - 7)`.
Differentiate the following w.r.t.x:
`sqrt(x^2 + sqrt(x^2 + 1)`
Differentiate the following w.r.t.x: `(8)/(3root(3)((2x^2 - 7x - 5)^11`
Differentiate the following w.r.t.x:
`(sqrt(3x - 5) - 1/sqrt(3x - 5))^5`
Differentiate the following w.r.t.x: `5^(sin^3x + 3)`
Differentiate the following w.r.t.x: `e^(3sin^2x - 2cos^2x)`
Differentiate the following w.r.t.x: cos2[log(x2 + 7)]
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 (sec 3x+ tan 3x)
Differentiate the following w.r.t.x:
`log(sqrt((1 - cos3x)/(1 + cos3x)))`
Differentiate the following w.r.t.x: `log(sqrt((1 - sinx)/(1 + sinx)))`
Differentiate the following w.r.t.x:
`log[a^(cosx)/((x^2 - 3)^3 logx)]`
Differentiate the following w.r.t. x : cot–1(x3)
Differentiate the following w.r.t. x : cot–1(4x)
Differentiate the following w.r.t. x : `cot^-1[cot(e^(x^2))]`
Differentiate the following w.r.t. x : `"cosec"^-1[1/cos(5^x)]`
Differentiate the following w.r.t. x : `"cosec"^-1((1)/(4cos^3 2x - 3cos2x))`
Differentiate the following w.r.t. x : `tan^-1(sqrt((1 + cosx)/(1 - cosx)))`
Differentiate the following w.r.t. x : `sin^-1((4sinx + 5cosx)/sqrt(41))`
Differentiate the following w.r.t. x :
`cos^-1((1 - x^2)/(1 + x^2))`
Differentiate the following w.r.t. x : `tan^-1((2x)/(1 - x^2))`
Differentiate the following w.r.t. x : `sin^-1(2xsqrt(1 - x^2))`
Differentiate the following w.r.t. x:
`tan^-1((2x^(5/2))/(1 - x^5))`
Differentiate the following w.r.t. x : `cot^-1((1 - sqrt(x))/(1 + sqrt(x)))`
Differentiate the following w.r.t. x : `tan^-1((8x)/(1 - 15x^2))`
Differentiate the following w.r.t. x : `root(3)((4x - 1)/((2x + 3)(5 - 2x)^2)`
Differentiate the following w.r.t. x : (sin x)x
Differentiate the following w.r.t. x: xe + xx + ex + ee.
Differentiate the following w.r.t. x:
`x^(x^x) + e^(x^x)`
Differentiate the following w.r.t. x : `x^(e^x) + (logx)^(sinx)`
Differentiate the following w.r.t. x :
etanx + (logx)tanx
Show that `bb("dy"/"dx" = y/x)` in the following, where a and p are constant:
xpy4 = (x + y)p+4, p ∈ N
Show that `"dy"/"dx" = y/x` in the following, where a and p are constants : `tan^-1((3x^2 - 4y^2)/(3x^2 + 4y^2))` = a2
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
Show that `"dy"/"dx" = y/x` in the following, where a and p are constants : `sin((x^3 - y^3)/(x^3 + y^3))` = a3
Differentiate y = etanx w.r. to x
If f(x) = 3x - 2 and g(x) = x2, then (fog)(x) = ________.
Derivative of (tanx)4 is ______
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 x2 + y2 - 2axy = 0, then `dy/dx` equals ______
If x = eθ, (sin θ – cos θ), y = eθ (sin θ + cos θ) then `dy/dx` at θ = `π/4` is ______.
If y = log (sec x + tan x), find `dy/dx`.
