Advertisements
Advertisements
Question
`"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`
Advertisements
Solution
y = `(sec^-1 "x")^2 ,"x" > 0`
⇒ `(d"y")/(d"x") = 2 sec^-1 "x"· (d(sec^-1"x"))/(d"x")`
⇒ `(d"y")/(d"x") = 2 sec^-1 "x"·(1)/(xsqrt(x^2 - 1))` ......(i)
⇒ `(d^2y)/(dx^2) = 2[1/(x^2(x^2 - 1))] + 2sec^-1x[[-sqrt(x^2 - 1) - x((2x)/(2sqrt(x^2 - 1))))/(x^2(x^2 - 1))]`
⇒ `(d^2"y")/(d"x"^2) = 2 [(1)/("x"^2("x"^2 -1)]] + 2 sec^-1 "x"· (1)/(xsqrt("x"^2 - 1)) [ ("x"(1 - 2"x"^2))/("x"^2 ("x"^2 - 1))] ` .......(ii)
From (i) and (ii), we get
`(d^2"y")/(d"x"^2) = 2 [(1)/("x"^2("x"^2 -1)]] + (d"y")/(d"x") [ ("x"(1 - 2"x"^2))/("x"^2 ("x"^2 - 1))] `
⇒ `"x"^2 ("x"^2 -1) (d^2"y")/(d"x"^2) + (2"x"^3 - "x")· (d"y")/(d"x") - 2 = 0`
APPEARS IN
RELATED QUESTIONS
Differentiate the function with respect to x.
cos (sin x)
Differentiate the function with respect to x.
cos x3 . sin2 (x5)
Differentiate the function with respect to x.
`2sqrt(cot(x^2))`
Differentiate the function with respect to x:
sin3 x + cos6 x
Differentiate the function with respect to x:
`(5x)^(3cos 2x)`
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`.
Does there exist a function which is continuos everywhere but not differentiable at exactly two points? Justify your answer?
Let f(x) = x|x|, for all x ∈ R. Discuss the derivability of f(x) at x = 0
If y = tanx + secx, prove that `("d"^2y)/("d"x^2) = cosx/(1 - sinx)^2`
| 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.
Show that the function f(x) = |sin x + cos x| is continuous at x = π.
`sin sqrt(x) + cos^2 sqrt(x)`
`cos(tan sqrt(x + 1))`
`sin^-1 1/sqrt(x + 1)`
`sec^-1 (1/(4x^3 - 3x)), 0 < x < 1/sqrt(2)`
For the curve `sqrt(x) + sqrt(y)` = 1, `"dy"/"dx"` at `(1/4, 1/4)` is ______.
The differential coefficient of `"tan"^-1 ((sqrt(1 + "x") - sqrt (1 - "x"))/(sqrt (1+ "x") + sqrt (1 - "x")))` 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
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 ______.
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 ______.
The function f(x) = x | x |, x ∈ R is differentiable ______.
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 ______.
Prove that the greatest integer function defined by f(x) = [x], 0 < x < 3 is not differentiable at x = 1 and x = 2.
