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.
sin (ax + b)
Differentiate the function with respect to x.
`(sin (ax + b))/cos (cx + d)`
Prove that the function f given by f(x) = |x − 1|, x ∈ R is not differentiable at x = 1.
Differentiate the function with respect to x:
(3x2 – 9x + 5)9
If f(x) = |x|3, show that f"(x) exists for all real x and find it.
Let f(x)= |cosx|. Then, ______.
`sin sqrt(x) + cos^2 sqrt(x)`
sinx2 + sin2x + sin2(x2)
`cos^-1 ((sinx + cosx)/sqrt(2)), (-pi)/4 < x < pi/4`
`tan^-1 (sqrt((1 - cosx)/(1 + cosx))), - pi/4 < x < pi/4`
`sec^-1 (1/(4x^3 - 3x)), 0 < x < 1/sqrt(2)`
If y = `sqrt(sinx + y)`, then `"dy"/"dx"` is equal to ______.
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 ____________.
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
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) = `{{:((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 ______.
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.
If \[u\] and \[v\] are differentiable functions, which formula is correct?
If a function \[f\] is differentiable at a point \[c\], what must be true at that point?
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?
Which conclusion establishes that \[f\] is continuous at \[x=c\]?
