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:
(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(e^((3x + 2) + 5)`
Differentiate the following w.r.t.x: cos2[log(x2 + 7)]
Differentiate the following w.r.t.x:
tan[cos(sinx)]
Differentiate the following w.r.t.x: `log[sec (e^(x^2))]`
Differentiate the following w.r.t.x: `(e^sqrt(x) + 1)/(e^sqrt(x) - 1)`
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 : `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 : `tan^-1[(1 - tan(x/2))/(1 + tan(x/2))]`
Differentiate the following w.r.t. x : `"cosec"^-1((1)/(4cos^3 2x - 3cos2x))`
Differentiate the following w.r.t. x : `tan^-1[(1 + cos(x/3))/(sin(x/3))]`
Differentiate the following w.r.t. x : `cot^-1((sin3x)/(1 + cos3x))`
Differentiate the following w.r.t. x : `tan^-1((cos7x)/(1 + sin7x))`
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)))]`
Differentiate the following w.r.t. x : `sin^-1((4sinx + 5cosx)/sqrt(41))`
Differentiate the following w.r.t. x : `sin^-1((cossqrt(x) + sinsqrt(x))/sqrt(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(2xsqrt(1 - x^2))`
Differentiate the following w.r.t. x : `tan^-1((2^x)/(1 + 2^(2x + 1)))`
Differentiate the following w.r.t. x : `cot^-1((4 - x - 2x^2)/(3x + 2))`
Differentiate the following w.r.t. x :
`(x + 1)^2/((x + 2)^3(x + 3)^4`
Differentiate the following w.r.t. x : `root(3)((4x - 1)/((2x + 3)(5 - 2x)^2)`
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
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 : `sec((x^5 + y^5)/(x^5 - y^5))` = 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
Differentiate y = `sqrt(x^2 + 5)` w.r. to x
Differentiate y = etanx w.r. to x
Differentiate `sin^-1((2cosx + 3sinx)/sqrt(13))` w.r. to x
If y = `sin^-1[("a"cosx - "b"sinx)/sqrt("a"^2 + "b"^2)]`, then find `("d"y)/("d"x)`
If f(x) = 3x - 2 and g(x) = x2, then (fog)(x) = ________.
If y = `1 + x + x^2/(2!) + x^3/(3!) + x^4/(4!) + .....,` then `(d^2y)/(dx^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 ______.
Differentiate `tan^-1 (sqrt((3 - x)/(3 + x)))` w.r.t. x.
