Advertisements
Advertisements
प्रश्न
Differentiate the following w.r.t. x : `sin^-1((4sinx + 5cosx)/sqrt(41))`
Advertisements
उत्तर
Let y = `sin^-1((4sinx + 5cosx)/sqrt(41))`
= `sin^-1[(sinx)((4)/(sqrt(41))) + (cosx)((5)/(sqrt(41)))]`
Since, `(4/sqrt(41))^2 + (5/sqrt(41))^2 = (16)/(41) + (25)/(41) = 1`,
we can write, `(4)/sqrt(41) = cos ∞ and (5)/sqrt(41) = sin ∞`.
∴ y = sin–1 (sin x cos ∞ + cos x sin ∞)
= sin–1[sin(x + ∞)]
= x + ∞, where ∞ is a constant
Differentiating w.r.t. x, we get
`"dy"/"dx" = "d"/"dx"(x + ∞)`
= `"d"/"dx"(x) + "d"/"dx"(∞)`
= 1 + 0
= 1.
APPEARS IN
संबंधित प्रश्न
Differentiate the following w.r.t. x:
(x3 – 2x – 1)5
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: log[cos(x3 – 5)]
Differentiate the following w.r.t.x: (1 + 4x)5 (3 + x −x2)8
Differentiate the following w.r.t.x: `(e^sqrt(x) + 1)/(e^sqrt(x) - 1)`
Differentiate the following w.r.t.x: `log[(ex^2(5 - 4x)^(3/2))/root(3)(7 - 6x)]`
Differentiate the following w.r.t.x:
`log[a^(cosx)/((x^2 - 3)^3 logx)]`
Differentiate the following w.r.t.x:
y = (25)log5(secx) − (16)log4(tanx)
Differentiate the following w.r.t. x:
`(x^2 + 2)^4/(sqrt(x^2 + 5)`
Differentiate the following w.r.t. x : tan–1(log x)
Differentiate the following w.r.t. x : cosec–1 (e–x)
Differentiate the following w.r.t. x : cot–1(4x)
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 : `sin^-1(x^(3/2))`
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 - cos(x^2))/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 : `tan^-1((2x)/(1 - x^2))`
Differentiate the following w.r.t. x : `sin^-1((1 - x^2)/(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(3x – 4x3)
Differentiate the following w.r.t. x : `cos^-1((e^x - e^(-x))/(e^x + e^(-x)))`
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 : `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 : `cot^-1((4 - x - 2x^2)/(3x + 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 : `(x^5.tan^3 4x)/(sin^2 3x)`
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
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 f(x) is odd and differentiable, then f′(x) is
Differentiate sin2 (sin−1(x2)) w.r. to x
Differentiate `cot^-1((cos x)/(1 + sinx))` w.r. to x
If y = `sin^-1[("a"cosx - "b"sinx)/sqrt("a"^2 + "b"^2)]`, then find `("d"y)/("d"x)`
If y = `1 + x + x^2/(2!) + x^3/(3!) + x^4/(4!) + .....,` then `(d^2y)/(dx^2)` = ______
The differential equation of the family of curves y = `"ae"^(2(x + "b"))` is ______.
Let f(x) be a polynomial function of the second degree. If f(1) = f(–1) and a1, a2, a3 are in AP, then f’(a1), f’(a2), f’(a3) are in ______.
If `cos((x^2 - y^2)/(x^2 + y^2))` = log a, show that `dy/dx = y/x`
