Advertisements
Advertisements
प्रश्न
`tan^-1 (secx + tanx), - pi/2 < x < pi/2`
Advertisements
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.
sin (x2 + 5)
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.
cos x3 . sin2 (x5)
Differentiate the function with respect to x.
`cos (sqrtx)`
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:
`sin^(–1)(xsqrtx), 0 ≤ x ≤ 1`
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 f(x) = x + 1, find `d/dx (fof) (x)`
Let f(x) = x|x|, for all x ∈ R. Discuss the derivability of f(x) at x = 0
If y = tan(x + y), find `("d"y)/("d"x)`
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 |
cos |x| is differentiable everywhere.
sinn (ax2 + bx + c)
sinx2 + sin2x + sin2(x2)
sinmx . cosnx
(x + 1)2(x + 2)3(x + 3)4
`sec^-1 (1/(4x^3 - 3x)), 0 < x < 1/sqrt(2)`
`tan^-1 ((sqrt(1 + x^2) + sqrt(1 - x^2))/(sqrt(1 + x^2) - sqrt(1 - x^2))), -1 < x < 1, x ≠ 0`
If xm . yn = (x + y)m+n, prove that `("d"^2"y")/("dx"^2)` = 0
For the curve `sqrt(x) + sqrt(y)` = 1, `"dy"/"dx"` at `(1/4, 1/4)` is ______.
If k be an integer, then `lim_("x" -> "k") ("x" - ["x"])` ____________.
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
If `ysqrt(1 - x^2) + xsqrt(1 - y^2)` = 1, then prove that `(dy)/(dx) = - sqrt((1 - y^2)/(1 - x^2))`
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 ______.
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 ______.
The function f(x) = x | x |, x ∈ R is differentiable ______.
