Advertisements
Advertisements
प्रश्न
Differentiate `sin^-1((2cosx + 3sinx)/sqrt(13))` w.r. to x
Advertisements
उत्तर
Let y = `sin^-1((2cosx + 3sinx)/sqrt(13))`
= `sin^-1((2cosx)/sqrt(13) + (3sinx)/sqrt(13))`
Put `2/sqrt(13)` = sin t and `3/sqrt(13)` = cos t
Also, sin2t + cos2t = `4/13 + 9/13` = 1
and tan t = `2/3`
∴ t = `tan^-1(2/3)`
∴ y = sin–1(sin t . cos x + cos t . sin x)
= sin–1[sin(t + x)]
= t + x
= `tan^-1(2/3) + x`
Differentiating w. r. t. x, we get
`("d"y)/("d"x) = "d"/("d"x)[tan^-1(2/3) + x]`
= 0 + 1
= 1
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: cos(x2 + a2)
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: `"cosec"(sqrt(cos x))`
Differentiate the following w.r.t.x: `e^(3sin^2x - 2cos^2x)`
Differentiate the following w.r.t.x: log[tan3x.sin4x.(x2 + 7)7]
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: `log[4^(2x)((x^2 + 5)/(sqrt(2x^3 - 4)))^(3/2)]`
Differentiate the following w.r.t.x:
`log[a^(cosx)/((x^2 - 3)^3 logx)]`
Differentiate the following w.r.t. x :
cos3[cos–1(x3)]
Differentiate the following w.r.t. x : `tan^-1[(1 + cos(x/3))/(sin(x/3))]`
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 : `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 :
`cos^-1 ((1 - 9^x))/((1 + 9^x)`
Differentiate the following w.r.t. x :
`sin^-1(4^(x + 1/2)/(1 + 2^(4x)))`
Differentiate the following w.r.t. x : `sin^-1 ((1 - 25x^2)/(1 + 25x^2))`
Differentiate the following w.r.t. x : `tan^-1((8x)/(1 - 15x^2))`
Differentiate the following w.r.t. x : `((x^2 + 2x + 2)^(3/2))/((sqrt(x) + 3)^3(cosx)^x`
Differentiate the following w.r.t. x : (logx)x – (cos x)cotx
Differentiate the following w.r.t. x : `x^(e^x) + (logx)^(sinx)`
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 `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: `log((x^20 - y^20)/(x^20 + y^20))` = 20
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 f(x) is odd and differentiable, then f′(x) is
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
If the function f(x) = `(log (1 + "ax") - log (1 - "bx))/x, x ≠ 0` is continuous at x = 0 then, f(0) = _____.
If `t = v^2/3`, then `(-v/2 (df)/dt)` is equal to, (where f is acceleration) ______
Derivative of (tanx)4 is ______
y = {x(x - 3)}2 increases for all values of x lying in the interval.
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 x2 + y2 - 2axy = 0, then `dy/dx` equals ______
If y = cosec x0, then `"dy"/"dx"` = ______.
Differentiate `tan^-1 (sqrt((3 - x)/(3 + x)))` w.r.t. x.
If `cos((x^2 - y^2)/(x^2 + y^2))` = log a, show that `dy/dx = y/x`
