Advertisements
Advertisements
प्रश्न
Differentiate the following w.r.t.x:
`log(sqrt((1 + cos((5x)/2))/(1 - cos((5x)/2))))`
Advertisements
उत्तर
Using `log(a/b)` = log a − log b
log ab = b log a
`y = log(sqrt(1 + cos ((5x)/2))) - log(sqrt(1 - cos ((5x)/2)))`
`y = log[1 + cos ((5x)/2)]^(1/2) - log[1 - cos((5x)/2)]^(1/2)`
`y = (1)/(2)log[1 + cos((5x)/2)] - (1)/(2)log[(1 - cos((5x)/2)]`
Differentiating w.r.t. x
`"dy"/"dx" = 1/2 × 1/(1 + cos((5x)/2)) × "d"/"dx"(1 + cos (5x)/2) - 1/2 × 1/(1 - cos((5x)/2)) × "d"/"dx"(1 - cos (5x)/(2))`
`"dy"/"dx" = 1/2 × 1/(1 + cos((5x)/2)) × [0 - sin ((5x)/2)] . 5/2 "d"/"dx" x - 1/2 × 1/(1 - cos((5x)/2)) × [0 + sin ((5x)/2)] . 5/2 "d"/"dx" x`
`"dy"/"dx" = 1/2 × 1/(1 + cos((5x)/2)) × - sin ((5x)/2) . 5/2 - 1/2 × 1/(1 - cos((5x)/2)) × sin ((5x)/2) . 5/2`
`"dy"/"dx" = [- 5sin((5x)/2)]/[4(1 + cos((5x)/2))] - [5sin((5x)/2)]/[4(1 - cos((5x)/2))]`
`"dy"/"dx" = [- 5sin((5x)/2)]/4. [1/(1 + cos((5x)/(2))) + 1/(1 - cos((5x)/(2)))]`
`"dy"/"dx" = [- 5sin((5x)/2)]/4. [(1 - cos ((5x)/2) + 1 + cos ((5x)/2)]/(1^2 - cos^2 ((5x)/2))]`
`"dy"/"dx" = [- 5sin((5x)/2)]/4. 2/(sin^2((5x)/2))` ...[ ∵ 1 – cos2x = sin2x]
`"dy"/"dx" = - 5/2 . 1/(sin((5x)/2))`
`"dy"/"dx" = - 5/2 . "cosec" ((5x)/2)`
APPEARS IN
संबंधित प्रश्न
Differentiate the following w.r.t.x:
`(2x^(3/2) - 3x^(4/3) - 5)^(5/2)`
Differentiate the following w.r.t.x: `log[tan(x/2)]`
Differentiate the following w.r.t.x: `sqrt(tansqrt(x)`
Differentiate the following w.r.t.x: log[cos(x3 – 5)]
Differentiate the following w.r.t.x: cos2[log(x2 + 7)]
Differentiate the following w.r.t.x: [log {log(logx)}]2
Differentiate the following w.r.t.x:
(x2 + 4x + 1)3 + (x3− 5x − 2)4
Differentiate the following w.r.t.x: (1 + sin2 x)2 (1 + cos2 x)3
Differentiate the following w.r.t.x:
`sqrt(cosx) + sqrt(cossqrt(x)`
Differentiate the following w.r.t.x:
log (sec 3x+ tan 3x)
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 : `tan^-1(sqrt(x))`
Differentiate the following w. r. t. x.
cos–1(1 – x2)
Differentiate the following w.r.t. x : `sin^4[sin^-1(sqrt(x))]`
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 : `tan^-1(sqrt((1 + cosx)/(1 - cosx)))`
Differentiate the following w.r.t.x:
tan–1 (cosec x + cot x)
Differentiate the following w.r.t. x : `sin^-1((4sinx + 5cosx)/sqrt(41))`
Differentiate the following w.r.t. x : `cos^-1((sqrt(3)cosx - sinx)/(2))`
Differentiate the following w.r.t. x : `"cosec"^-1[(10)/(6sin(2^x) - 8cos(2^x))]`
Differentiate the following w.r.t. x :
`cos^-1((1 - x^2)/(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 :
`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 : `cot^-1((1 - sqrt(x))/(1 + sqrt(x)))`
Differentiate the following w.r.t. x : `tan^-1((2sqrt(x))/(1 + 3x))`
Differentiate the following w.r.t. x : `cot^-1((a^2 - 6x^2)/(5ax))`
Differentiate the following w.r.t. x : `root(3)((4x - 1)/((2x + 3)(5 - 2x)^2)`
Differentiate the following w.r.t. x : `(x^2 + 3)^(3/2).sin^3 2x.2^(x^2)`
Differentiate the following w.r.t. x :
(sin x)tanx + (cos x)cotx
Differentiate y = `sqrt(x^2 + 5)` w.r. to x
Differentiate y = etanx w.r. to x
If y = `"e"^(1 + logx)` then find `("d"y)/("d"x)`
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 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) ______
The differential equation of the family of curves y = `"ae"^(2(x + "b"))` is ______.
Solve `x + y (dy)/(dx) = sec(x^2 + y^2)`
The value of `d/(dx)[tan^-1((a - x)/(1 + ax))]` 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`
