Advertisements
Advertisements
प्रश्न
Differentiate the following w.r.t. x :
`cot^-1[(sqrt(1 + sin ((4x)/3)) + sqrt(1 - sin ((4x)/3)))/(sqrt(1 + sin ((4x)/3)) - sqrt(1 - sin ((4x)/3)))]`
Advertisements
उत्तर
Let y = `cot^-1[(sqrt(1 + sin ((4x)/3)) + sqrt(1 - sin ((4x)/3)))/(sqrt(1 + sin ((4x)/3)) - sqrt(1 - sin ((4x)/3)))]`
= `1 + sin ((4x)/3)`
= `1 + cos(pi/2 - (4x)/3)`
= `2cos^2(pi/4 - (2x)/3)`
∴ `sqrt(1 + sin((4x)/3)) = sqrt(2)cos(pi/4 - (2x)/3)`
Also, `1 - sin ((4x)/3)`
= `1 - cos(pi/2 - (4x)/3)`
= `2sin^2(pi/4 - (2x)/3)`
∴ `sqrt(1 - sin((4x)/3)) = sqrt(2)sin(pi/4 - (2x)/3)`
∴ `(sqrt(1 + sin ((4x)/3)) + sqrt(1 - sin ((4x)/3)))/(sqrt(1 + sin((4x)/3) - sqrt(1 - sin((4x)/3)`
= `(sqrt(2)cos(pi/4 - (2x)/3) + sqrt(2)sin(pi/4 - (2x)/3))/(sqrt(2)cos(pi/4 - (2x)/3) - sqrt(2)sin(pi/4 - (2x)/3)`
= `(cos(pi/4 - (2x)/3) + sin(pi/4 - (2x)/3))/(cos(pi/4 - (2x)/3) - sin(pi/4 - (2x)/3)`
= `(1 + tan(pi/4 - (2x)/3))/(1 - tan(pi/4 - (2x)/3)) ...["Dividing by" cos(pi/4 - (2x)/3)`
= `(tan pi/4 + tan(pi/4 - (2x)/3))/(1 - tan pi/4. tan(pi/4 - (2x)/3)) ...[∵ tan pi/4 = 1]`
= `tan[pi/4 + pi/4 - (2x)/3]`
= `tan(pi/2 - (2x)/3)`
= `cot((2x)/3)`
∴ y = `cot^-1[cot((2x)/3)] = (2x)/(3)`
Differentiating w.r.t. x, we get
`"dy"/"dx" = "d"/"dx"((2x)/3)`
= `(2)/(3)"d"/"dx"(x)`
= `(2)/(3) xx 1`
= `(2)/(3)`
APPEARS IN
संबंधित प्रश्न
Differentiate the following w.r.t. x:
(x3 – 2x – 1)5
Differentiate the following w.r.t.x:
`(2x^(3/2) - 3x^(4/3) - 5)^(5/2)`
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:
`sqrt(e^((3x + 2) + 5)`
Differentiate the following w.r.t.x: cot3[log(x3)]
Differentiate the following w.r.t.x:
tan[cos(sinx)]
Differentiate the following w.r.t.x:
(x2 + 4x + 1)3 + (x3− 5x − 2)4
Differentiate the following w.r.t.x:
`(e^(2x) - e^(-2x))/(e^(2x) + e^(-2x))`
Differentiate the following w.r.t.x:
`log(sqrt((1 + cos((5x)/2))/(1 - cos((5x)/2))))`
Differentiate the following w.r.t.x: `log(sqrt((1 - sinx)/(1 + sinx)))`
Differentiate the following w.r.t. x : cot–1(x3)
Differentiate the following w.r.t. x : `tan^-1(sqrt(x))`
Differentiate the following w.r.t. x :
`sin^-1(sqrt((1 + x^2)/2))`
Differentiate the following w. r. t. x.
cos–1(1 – x2)
Differentiate the following w.r.t. x : `cot^-1((sin3x)/(1 + cos3x))`
Differentiate the following w.r.t. x : `cos^-1((sqrt(3)cosx - sinx)/(2))`
Differentiate the following w.r.t. x : `cos^-1((3cos3x - 4sin3x)/5)`
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(2xsqrt(1 - x^2))`
Differentiate the following w.r.t. x : `tan^-1((8x)/(1 - 15x^2))`
Differentiate the following w.r.t. x :
`tan^(−1)[(2^(x + 2))/(1 − 3(4^x))]`
Differentiate the following w.r.t. x : `tan^-1((a + btanx)/(b - atanx))`
Differentiate the following w.r.t. x : `cot^-1((4 - x - 2x^2)/(3x + 2))`
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 xx)
Differentiate the following w.r.t. x :
etanx + (logx)tanx
Differentiate the following w.r.t. x :
(sin x)tanx + (cos x)cotx
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 : `sin((x^3 - y^3)/(x^3 + y^3))` = a3
If y is a function of x and log (x + y) = 2xy, then the value of y'(0) = ______.
Solve the following :
The values of f(x), g(x), f'(x) and g'(x) are given in the following table :
| x | f(x) | g(x) | f'(x) | fg'(x) |
| – 1 | 3 | 2 | – 3 | 4 |
| 2 | 2 | – 1 | – 5 | – 4 |
Match the following :
| A Group – Function | B Group – Derivative |
| (A)`"d"/"dx"[f(g(x))]"at" x = -1` | 1. – 16 |
| (B)`"d"/"dx"[g(f(x) - 1)]"at" x = -1` | 2. 20 |
| (C)`"d"/"dx"[f(f(x) - 3)]"at" x = 2` | 3. – 20 |
| (D)`"d"/"dx"[g(g(x))]"at"x = 2` | 5. 12 |
Differentiate `cot^-1((cos x)/(1 + sinx))` w.r. to x
If x = `sqrt("a"^(sin^-1 "t")), "y" = sqrt("a"^(cos^-1 "t")), "then" "dy"/"dx"` = ______
If `t = v^2/3`, then `(-v/2 (df)/dt)` is equal to, (where f is acceleration) ______
y = {x(x - 3)}2 increases for all values of x lying in the interval.
A particle moves so that x = 2 + 27t - t3. The direction of motion reverses after moving a distance of ______ units.
If y = cosec x0, then `"dy"/"dx"` = ______.
If x = p sin θ, y = q cos θ, then `dy/dx` = ______
Solve `x + y (dy)/(dx) = sec(x^2 + y^2)`
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.
