Advertisements
Advertisements
प्रश्न
`tan^-1 (secx + tanx), - pi/2 < x < pi/2`
उत्तर
Let y = tan–1(sec x + tan x)
Differentiating both sides w.r.t. x
`"dy"/"dx" = "d"/"dx" [tan^-1 (secx + tanx)]`
= `1/(1 + (secx + tanx)^2) * "d"/"dx"(secx + tanx)`
= `1/(1 + sec^2 + tan^2x + 2 sec x tanx) * (secx tanx + sec^2x)`
= `1/((1 + tan^2x) + sec^2x + 2secx tanx) * secx(tanx + secx)`
= `1/(sec^2x + sec^2x + 2secx tanx) * secx(tanx + secx)`
= `1/(2sec^2x + 2secx tanx) * secx(tanx + secx)`
= `1/(2secx(secx + tanx)) * secx(tanx + secx)`
= `1/2`
Hence, `"dy"/"dx" = 1/2`
Alternative solution:
Let y = `tan^-1 (secx + tanx), (-pi)/2 < x < pi/2`
= `tan^-1 (1/cosx + sinx/cosx)`
= `tan^-1 ((1 + sinx)/cosx)`
= `tan^-1 [(cos^2 x/2 + sin^2 x/2 + 2sin x/2 cos x/2)/(cos^2 x/2 - sin^2 x/2)]` ......`[(because 2x = 2sinx cosx),(cos2x = cos^2x - sin^2x)]`
= `tan^-1 [(cos x/2 + sin x/2)^2/((cos x/2 + sin x/2)(cos x/2 - sin x/2))]`
= `tan^-1 [(cos x/2 + sin x/2)/(cos x/2 - sin x/2)]`
= `tan^-1 [(1 + tan x/2)/(1 - tan x/2)]` .....[Dividing the Nr. and Den. by cos `x/2`]
= `tan^-1 [(tan pi/4 + tan x/2),(1 - tan pi/4 * tan x/2)]`
= `tan^-1 [tan (pi/4 + x/2)]`
∴ y = `pi/4 + x/2`
Differentiating both sides w.r.t. x
`"dy"/"dx" = 1/2 "d"/"dx" (x)`
= `1/2 * 1`
= `1/2`
Hence, `"dy"/"dx" = 1/2`.
APPEARS IN
संबंधित प्रश्न
Differentiate the function with respect to x.
cos (sin x)
Differentiate the function with respect to x.
`cos (sqrtx)`
Differentiate w.r.t. x the function:
`sin^(–1)(xsqrtx ), 0 ≤ x ≤ 1`
Differentiate w.r.t. x the function:
`(cos^(-1) x/2)/sqrt(2x+7), -2 < x < 2`
If (x – a)2 + (y – b)2 = c2, for some c > 0, prove that `[1+ (dy/dx)^2]^(3/2)/((d^2y)/dx^2)` is a constant independent of a and b.
Does there exist a function which is continuos everywhere but not differentiable at exactly two points? Justify your answer?
If f(x) = x + 1, find `d/dx (fof) (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 ______.
cos |x| is differentiable everywhere.
`sin sqrt(x) + cos^2 sqrt(x)`
sinn (ax2 + bx + c)
(x + 1)2(x + 2)3(x + 3)4
`tan^-1 (sqrt((1 - cosx)/(1 + cosx))), - pi/4 < x < pi/4`
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 ____________.
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
If sin y = x sin (a + y), then value of dy/dx is
If `ysqrt(1 - x^2) + xsqrt(1 - y^2)` = 1, then prove that `(dy)/(dx) = - sqrt((1 - y^2)/(1 - x^2))`
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 ______.
The function f(x) = x | x |, x ∈ R is differentiable ______.
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.
