Advertisements
Advertisements
प्रश्न
Differentiate `tan^-1 (sqrt(1 - x^2)/x)` with respect to`cos^-1(2xsqrt(1 - x^2))`, where `x ∈ (1/sqrt(2), 1)`
Advertisements
उत्तर
Let u = `tan^-1 (sqrt(1 - x^2)/x)` and v = `cos^-1(2xsqrt(1 - x^2))`.
We want to find `"du"/"dv" = (("du")/("dx"))/(("dv")/("dx"))`
Now u = `tan^-1 (sqrt(1 - x^2)/x)`.
Put x = `sintheta. (pi/2 < theta < pi/2)`
Then u = `tan^-1 (sqrt(1 - sin^2theta)/sintheta)`
= `tan^-1 (cot theta)`
= `tan^-1 {tan (pi/2 - theta)}`
= `pi/2 - theta`
= `pi/2 - sin^-1x`
Hence `"du"/"dx" = (-1)/sqrt(1 - x^2)`.
Now v = `cos^-1 (2x sqrt(1 - x^2))`
= `pi/2 - sin^-1 (2x sqrt(1 - x^2))`
= `pi/2 - sin^-1 (2sintheta sqrt(1 - sin^2theta))`
= `pi/2 - sin^-1 (sin 2theta)`
= `pi/2 - sin^-1 {sin (pi - 2theta)}` .......{Since `pi/2` < 2θ < π]
= `pi/2 - (pi / 2theta)`
= `(-pi)/2 + 2theta`
⇒ v = `(-pi)/2 + 2sin^-1x`
⇒ `"dv"/"dv" = (("du")/("d"x))/(("dv")/("dx"))`
= `((-1)/sqrt(1 - x^2))/(2/sqrt(1 - x^2))`
= `(-1)/2`
APPEARS IN
संबंधित प्रश्न
Differentiate the function with respect to x.
cos (sin x)
Differentiate the function with respect to x.
sin (ax + b)
Differentiate the function with respect to x:
`sin^(–1)(xsqrtx), 0 ≤ x ≤ 1`
Differentiate the function with respect to x:
`(cos^(-1) x/2)/sqrt(2x+7)`, −2 < x < 2
Differentiate the function with respect to x:
`x^(x^2 -3) + (x -3)^(x^2)`, for x > 3
If f(x) = |x|3, show that f"(x) exists for all real x and find it.
If sin y = xsin(a + y) prove that `(dy)/(dx) = sin^2(a + y)/sin a`
If y = tan(x + y), find `("d"y)/("d"x)`
Let f(x)= |cosx|. Then, ______.
If u = `sin^-1 ((2x)/(1 + x^2))` and v = `tan^-1 ((2x)/(1 - x^2))`, then `"du"/"dv"` is ______.
| COLUMN-I | COLUMN-II |
| (A) If a function f(x) = `{((sin3x)/x, "if" x = 0),("k"/2",", "if" x = 0):}` is continuous at x = 0, then k is equal to |
(a) |x| |
| (B) Every continuous function is differentiable | (b) True |
| (C) An example of a function which is continuous everywhere but not differentiable at exactly one point |
(c) 6 |
| (D) The identity function i.e. f (x) = x ∀ ∈x R is a continuous function |
(d) False |
|sinx| is a differentiable function for every value of x.
cos |x| is differentiable everywhere.
`sin sqrt(x) + cos^2 sqrt(x)`
`sin^-1 1/sqrt(x + 1)`
`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 ((sqrt(1 + x^2) + sqrt(1 - x^2))/(sqrt(1 + x^2) - sqrt(1 - x^2))), -1 < x < 1, x ≠ 0`
If k be an integer, then `lim_("x" -> "k") ("x" - ["x"])` ____________.
The differential coefficient of `"tan"^-1 ((sqrt(1 + "x") - sqrt (1 - "x"))/(sqrt (1+ "x") + sqrt (1 - "x")))` is ____________.
A function is said to be continuous for x ∈ R, if ____________.
If sin y = x sin (a + y), then value of dy/dx is
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 ______.
If f(x) = `{{:((sin(p + 1)x + sinx)/x,",", x < 0),(q,",", x = 0),((sqrt(x + x^2) - sqrt(x))/(x^(3//2)),",", x > 0):}`
is continuous at x = 0, then the ordered pair (p, q) is equal to ______.
Let S = {t ∈ R : f(x) = |x – π| (e|x| – 1)sin |x| is not differentiable at t}. Then the set S is equal to ______.
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.
The function f(x) = x | x |, x ∈ R is differentiable ______.
If \[u\] and \[v\] are differentiable functions, which formula is correct?
If \[u\] and \[v\] are differentiable functions, what is \[(uv)'\]?
A function is differentiable at \[x=c\] when which condition holds?
Which limit is the left-hand derivative at \[x=c\]?
If \[f\] is differentiable at \[c\], which limit equals \[f'(c)\]?
For \[x\ne c\], which identity is used to prove that differentiability implies continuity?
Why is \[|x|\] not differentiable at \[x=0\]?
What does differentiability at a point mean?
