Advertisements
Advertisements
प्रश्न
Differentiate w.r.t. x the function:
`(cos^(-1) x/2)/sqrt(2x+7), -2 < x < 2`
उत्तर
Let, y = `(cos^-1 x/2)/(sqrt(2x + 7)) = u/v`
`therefore u = cos^-1 x/2, v = sqrt(2x + 7)`
Now, u = `cos^-1 x/2`
On differentiating with respect to x,
`(du)/dx = d/dx cos^-1 x/2`
`= - 1/(sqrt(1 - x^2/4)) d/dx (x/2)`
`= - 2/(sqrt(4 - x^2)) * 1/2`
`= (-1)/sqrt(4 - x^2)` ...(1)
and v = `sqrt(2x + 7)`
On differentiating with respect to x,
`(dv)/dx = 1/2 (2x - 7)^(1/2 - 1) d/dx (2x - 7)`
`= 1/2 (2x - 7)^(- 1//2) (2) = 1/(sqrt(2x + 7))` ...(2)
y = `u/v`
∴ `dy/dx = (v (du)/dx - u (dv)/dx)/v^2` ... [(1) and (2) substituting the value of]
`= (- 1/(sqrt(4 - x^2)) xx sqrt(2x + 7) - (cos^-1 x/2)/sqrt(2x + 7))/((2x + 7))`
`= - [1/(sqrt(4 - x^2) sqrt(2x + 7)) + (cos^-1 x/2)/(2x + 7)^(3//2)]`
APPEARS IN
संबंधित प्रश्न
Differentiate the function with respect to x.
sin (x2 + 5)
Differentiate the function with respect to x.
sin (ax + b)
Differentiate the function with respect to x.
`sec(tan (sqrtx))`
Differentiate the function with respect to x.
`cos (sqrtx)`
Differentiate w.r.t. x the function:
(3x2 – 9x + 5)9
Differentiate w.r.t. x the function:
`(5x)^(3cos 2x)`
Differentiate w.r.t. x the function:
`x^(x^2 -3) + (x -3)^(x^2)`, for x > 3
Find `dy/dx, if y = 12 (1 – cos t), x = 10 (t – sin t), -pi/2< t< pi/2`
`"If y" = (sec^-1 "x")^2 , "x" > 0 "show that" "x"^2 ("x"^2 - 1) (d^2"y")/(d"x"^2) + (2"x"^3 - "x") (d"y")/(d"x") - 2 = 0`
If y = tanx + secx, prove that `("d"^2y)/("d"x^2) = cosx/(1 - sinx)^2`
Let f(x)= |cosx|. Then, ______.
Differential coefficient of sec (tan–1x) w.r.t. x is ______.
If u = `sin^-1 ((2x)/(1 + x^2))` and v = `tan^-1 ((2x)/(1 - x^2))`, then `"du"/"dv"` is ______.
cos |x| is differentiable everywhere.
Show that the function f(x) = |sin x + cos x| is continuous at x = π.
(sin x)cosx
(x + 1)2(x + 2)3(x + 3)4
`cos^-1 ((sinx + cosx)/sqrt(2)), (-pi)/4 < x < pi/4`
`tan^-1 (sqrt((1 - cosx)/(1 + cosx))), - pi/4 < x < pi/4`
`tan^-1 (secx + tanx), - pi/2 < x < pi/2`
`tan^-1 (("a"cosx - "b"sinx)/("b"cosx - "a"sinx)), - pi/2 < x < pi/2` and `"a"/"b" tan x > -1`
If xm . yn = (x + y)m+n, prove that `("d"^2"y")/("dx"^2)` = 0
If y = `sqrt(sinx + y)`, then `"dy"/"dx"` is equal to ______.
For the curve `sqrt(x) + sqrt(y)` = 1, `"dy"/"dx"` at `(1/4, 1/4)` is ______.
If k be an integer, then `lim_("x" -> "k") ("x" - ["x"])` ____________.
A function is said to be continuous for x ∈ R, if ____________.
The rate of increase of bacteria in a certain culture is proportional to the number present. If it doubles in 5 hours then in 25 hours, its number would be
`d/(dx)[sin^-1(xsqrt(1 - x) - sqrt(x)sqrt(1 - x^2))]` is equal to
Let c, k ∈ R. If f(x) = (c + 1)x2 + (1 – c2)x + 2k and f(x + y) = f(x) + f(y) – xy, for all x, y ∈ R, then the value of |2(f(1) + f(2) + f(3) + ... + f(20))| is equal to ______.
A particle is moving on a line, where its position S in meters is a function of time t in seconds given by S = t3 + at2 + bt + c where a, b, c are constant. It is known that at t = 1 seconds, the position of the particle is given by S = 7 m. Velocity is 7 m/s and acceleration is 12 m/s2. The values of a, b, c are ______.
If f(x) = `{{:(ax + b; 0 < x ≤ 1),(2x^2 - x; 1 < x < 2):}` is a differentiable function in (0, 2), then find the values of a and b.
If f(x) = | cos x |, then `f((3π)/4)` is ______.
The set of all points where the function f(x) = x + |x| is differentiable, is ______.
