Advertisements
Advertisements
Question
`sec^-1 (1/(4x^3 - 3x)), 0 < x < 1/sqrt(2)`
Advertisements
Solution
Let y = `sec^-1 (1/(4x^3 - 3x))`
Put x = cos θ
∴ θ = `cos^-1x`
y = `sec^-1 (1/(4cos^3theta - 3 cos theta))`
⇒ y = `sec^-1 (1/(cos 3theta))` .....[∵ cos 3θ = 4 cos3θ – 3 cos θ]
⇒ y = `sec^-1 (sec 3theta)`
⇒ y = 3θ
y = `3cos^-1x`
Differentiating both sides w.r.t. x
`"dy"/"d" = 3 * "d"/"dx" cos^-1x`
= `3((-1)/sqrt(1 - x^2))`
= `(-3)/sqrt(1 - x^2)`
Hence, `"dy"/"dx" = (-3)/sqrt(1 - x^2)`.
APPEARS IN
RELATED QUESTIONS
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 (ax + b))/cos (cx + d)`
Differentiate the function with respect to x.
`2sqrt(cot(x^2))`
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
Find `dy/dx`, if y = 12 (1 – cos t), x = 10 (t – sin t), `-pi/2 < t < pi/2`.
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 f(x) = x + 1, find `d/dx (fof) (x)`
If y = tan(x + y), find `("d"y)/("d"x)`
Differentiate `tan^-1 (sqrt(1 - x^2)/x)` with respect to`cos^-1(2xsqrt(1 - x^2))`, where `x ∈ (1/sqrt(2), 1)`
| 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^-1 1/sqrt(x + 1)`
sinmx . cosnx
`tan^-1 (secx + tanx), - pi/2 < x < pi/2`
For the curve `sqrt(x) + sqrt(y)` = 1, `"dy"/"dx"` at `(1/4, 1/4)` is ______.
If `"f"("x") = ("sin" ("e"^("x"-2) - 1))/("log" ("x" - 1)), "x" ne 2 and "f" ("x") = "k"` for x = 2, then value of k for which f is continuous is ____________.
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 ______.
Let f: R→R and f be a differentiable function such that f(x + 2y) = f(x) + 4f(y) + 2y(2x – 1) ∀ x, y ∈ R and f’(0) = 1, then f(3) + f’(3) is ______.
Prove that the greatest integer function defined by f(x) = [x], 0 < x < 3 is not differentiable at x = 1 and x = 2.
Differentiability determines whether a function has what at a particular point?
Which expression defines the derivative of a real function \[f\] at a point \[c\] in its domain?
What is \[\frac{d}{dx}(\sin x)\]?
What is \[\frac{d}{dx}(\tan x)\]?
Which limit is the right-hand derivative at \[x=c\]?
What is the practical condition for differentiability at a point?
Which conclusion establishes that \[f\] is continuous at \[x=c\]?
What does differentiability at a point mean?
