Advertisements
Advertisements
Question
Differentiate the following w.r.t. x : `tan^-1((2^x)/(1 + 2^(2x + 1)))`
Advertisements
Solution
Let y = `tan^-1((2^x)/(1 + 2^(2x + 1)))`
= `tan^-1[(2.2^x - 2^x)/(1 + (2.2^x)(2^x))]`
= tan–1(2.2x) – tan–1(2x)
Differentiating w.r.t. x, we get
`"dy"/"dx" = "d"/"dx"[tan^-1(2.2^x) - tan^-1(2^x)]`
= `"d"/"dx"[tan^-1(2.2^x)] - "d"/"dx"[tan^-1(2^x)]`
= `(1)/(1 + (2.2^x)^2)."d"/"dx"(2.2^x) - (1)/(1 + (2^x)^2)."d"/"dx"(2^x)`
= `(1)/(1 + 4(2^(2x))) xx 2 xx 2^xlog2 - (1)/(1 + 2^(2x)) xx 2^xlog2`
= `2^xlog2[2/(1 + 4(2^(2x))) - 1/(1 + 2^(2x))]`.
APPEARS IN
RELATED QUESTIONS
Differentiate the following w.r.t.x:
`sqrt(x^2 + sqrt(x^2 + 1)`
Differentiate the following w.r.t.x: cos(x2 + a2)
Differentiate the following w.r.t.x: `5^(sin^3x + 3)`
Differentiate the following w.r.t.x: `e^(3sin^2x - 2cos^2x)`
Differentiate the following w.r.t.x: `log[sec (e^(x^2))]`
Differentiate the following w.r.t.x: `cot(logx/2) - log(cotx/2)`
Differentiate the following w.r.t.x:
`log(sqrt((1 - cos3x)/(1 + cos3x)))`
Differentiate the following w.r.t.x:
`log(sqrt((1 + cos((5x)/2))/(1 - cos((5x)/2))))`
Differentiate the following w.r.t.x:
`log[a^(cosx)/((x^2 - 3)^3 logx)]`
Differentiate the following w.r.t. x : tan–1(log x)
Differentiate the following w.r.t. x :
`sin^-1(sqrt((1 + x^2)/2))`
Differentiate the following w.r.t. x : `cos^-1(sqrt((1 + cosx)/2))`
Differentiate the following w.r.t. x :
`cos^-1(sqrt(1 - cos(x^2))/2)`
Differentiate the following w.r.t. x : `tan^-1((cos7x)/(1 + sin7x))`
Differentiate the following w.r.t. x : `sin^-1((4sinx + 5cosx)/sqrt(41))`
Differentiate the following w.r.t. x : `cos^-1((3cos3x - 4sin3x)/5)`
Differentiate the following w.r.t. x : `cos^-1((e^x - e^(-x))/(e^x + e^(-x)))`
Differentiate the following w.r.t. x :
`sin^(−1) ((1 − x^3)/(1 + x^3))`
Differentiate the following w.r.t. x : `cot^-1((1 - sqrt(x))/(1 + sqrt(x)))`
Differentiate the following w.r.t. x : `tan^-1((8x)/(1 - 15x^2))`
Differentiate the following w.r.t.x:
`cot^-1((1 + 35x^2)/(2x))`
Differentiate the following w.r.t. x : `cot^-1((a^2 - 6x^2)/(5ax))`
Differentiate the following w.r.t. x : (sin x)x
Differentiate the following w.r.t. x: xe + xx + ex + ee.
Differentiate the following w.r.t. x : `x^(e^x) + (logx)^(sinx)`
Differentiate the following w.r.t. x :
(sin x)tanx + (cos x)cotx
Differentiate the following w.r.t. x : `[(tanx)^(tanx)]^(tanx) "at" x = pi/(4)`
Show that `bb("dy"/"dx" = y/x)` in the following, where a and p are constant:
xpy4 = (x + y)p+4, p ∈ N
Show that `"dy"/"dx" = y/x` in the following, where a and p are constants : `sin((x^3 - y^3)/(x^3 + y^3))` = a3
If y = `"e"^(1 + logx)` then find `("d"y)/("d"x)`
Differentiate sin2 (sin−1(x2)) w.r. to x
Differentiate `tan^-1((8x)/(1 - 15x^2))` w.r. to x
If y = `sin^-1[("a"cosx - "b"sinx)/sqrt("a"^2 + "b"^2)]`, then find `("d"y)/("d"x)`
If f(x) = 3x - 2 and g(x) = x2, then (fog)(x) = ________.
If y = `(3x^2 - 4x + 7.5)^4, "then" dy/dx` is ______
The differential equation of the family of curves y = `"ae"^(2(x + "b"))` is ______.
If x2 + y2 - 2axy = 0, then `dy/dx` equals ______
If y = cosec x0, then `"dy"/"dx"` = ______.
Solve `x + y (dy)/(dx) = sec(x^2 + y^2)`
Find `(dy)/(dx)`, if x3 + x2y + xy2 + y3 = 81
The value of `d/(dx)[tan^-1((a - x)/(1 + ax))]` is ______.
Let f(x) be a polynomial function of the second degree. If f(1) = f(–1) and a1, a2, a3 are in AP, then f’(a1), f’(a2), f’(a3) are in ______.
Differentiate `tan^-1 (sqrt((3 - x)/(3 + x)))` w.r.t. x.
If `cos((x^2 - y^2)/(x^2 + y^2))` = log a, show that `dy/dx = y/x`
