Advertisements
Advertisements
Question
Differentiate the following w.r.t.x:
`log(sqrt((1 + cos((5x)/2))/(1 - cos((5x)/2))))`
Advertisements
Solution
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
RELATED QUESTIONS
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:
sin2x2 – cos2x2
Differentiate the following w.r.t.x: (1 + 4x)5 (3 + x −x2)8
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(sqrt((1 - cos3x)/(1 + cos3x)))`
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 : cosec–1 (e–x)
Differentiate the following w.r.t. x : cot–1(x3)
Differentiate the following w.r.t. x : `tan^-1(sqrt(x))`
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 :
`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((cossqrt(x) + sinsqrt(x))/sqrt(2))`
Differentiate the following w.r.t. x : `cos^-1((3cos3x - 4sin3x)/5)`
Differentiate the following w.r.t. x : `"cosec"^-1[(10)/(6sin(2^x) - 8cos(2^x))]`
Differentiate the following w.r.t. x : `sin^-1((1 - x^2)/(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:
`tan^-1((2x^(5/2))/(1 - x^5))`
Differentiate the following w.r.t. x : `tan^-1((8x)/(1 - 15x^2))`
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 + 2x + 2)^(3/2))/((sqrt(x) + 3)^3(cosx)^x`
Differentiate the following w.r.t. x: `x^(tan^(-1)x`
Differentiate the following w.r.t. x : (logx)x – (cos x)cotx
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 : `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 : `cos^-1((7x^4 + 5y^4)/(7x^4 - 5y^4)) = tan^-1a`
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) = ______.
Differentiate y = etanx w.r. to x
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 = `sqrt(cos x + sqrt(cos x + sqrt(cos x + ...... ∞)`, show that `("d"y)/("d"x) = (sin x)/(1 - 2y)`
If the function f(x) = `(log (1 + "ax") - log (1 - "bx))/x, x ≠ 0` is continuous at x = 0 then, f(0) = _____.
Derivative of (tanx)4 is ______
If y = `(3x^2 - 4x + 7.5)^4, "then" dy/dx` is ______
If x2 + y2 - 2axy = 0, then `dy/dx` equals ______
The volume of a spherical balloon is increasing at the rate of 10 cubic centimetre per minute. The rate of change of the surface of the balloon at the instant when its radius is 4 centimetres, is ______
Solve `x + y (dy)/(dx) = sec(x^2 + y^2)`
Differentiate `tan^-1 (sqrt((3 - x)/(3 + x)))` w.r.t. 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`.
