Advertisements
Advertisements
Question
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
Solution
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
RELATED QUESTIONS
Differentiate the following w.r.t.x: cos(x2 + a2)
Differentiate the following w.r.t.x: `"cosec"(sqrt(cos x))`
Differentiate the following w.r.t.x: cos2[log(x2 + 7)]
Differentiate the following w.r.t.x: `log[sec (e^(x^2))]`
Differentiate the following w.r.t.x: `log_(e^2) (log x)`
Differentiate the following w.r.t.x:
(x2 + 4x + 1)3 + (x3− 5x − 2)4
Differentiate the following w.r.t.x:
`(x^3 - 5)^5/(x^3 + 3)^3`
Differentiate the following w.r.t.x: `(1 + sinx°)/(1 - sinx°)`
Differentiate the following w.r.t.x:
`(e^(2x) - e^(-2x))/(e^(2x) + e^(-2x))`
Differentiate the following w.r.t.x: `log[4^(2x)((x^2 + 5)/(sqrt(2x^3 - 4)))^(3/2)]`
Differentiate the following w.r.t. x : cosec–1 (e–x)
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 :
cos3[cos–1(x3)]
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 : `cos^-1(sqrt((1 + cosx)/2))`
Differentiate the following w.r.t. x : `tan^-1[(1 - tan(x/2))/(1 + tan(x/2))]`
Differentiate the following w.r.t. x : `cos^-1((sqrt(3)cosx - sinx)/(2))`
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 : `cos^-1((e^x - e^(-x))/(e^x + e^(-x)))`
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: xe + xx + ex + ee.
Differentiate the following w.r.t. x : (logx)x – (cos x)cotx
Differentiate the following w.r.t. x :
(sin x)tanx + (cos x)cotx
Differentiate the following w.r.t. x : `[(tanx)^(tanx)]^(tanx) "at" x = pi/(4)`
Show that `"dy"/"dx" = y/x` in the following, where a and p are constants : x7.y5 = (x + y)12
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 |
If y = `tan^-1[sqrt((1 + cos x)/(1 - cos x))]`, find `("d"y)/("d"x)`
Differentiate `cot^-1((cos x)/(1 + sinx))` w.r. to x
Differentiate `sin^-1((2cosx + 3sinx)/sqrt(13))` w.r. to x
If y = `sqrt(cos x + sqrt(cos x + sqrt(cos x + ...... ∞)`, show that `("d"y)/("d"x) = (sin x)/(1 - 2y)`
Derivative of (tanx)4 is ______
The weight W of a certain stock of fish is given by W = nw, where n is the size of stock and w is the average weight of a fish. If n and w change with time t as n = 2t2 + 3 and w = t2 - t + 2, then the rate of change of W with respect to t at t = 1 is ______
The differential equation of the family of curves y = `"ae"^(2(x + "b"))` 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.
If y = log (sec x + tan x), find `dy/dx`.
