Advertisements
Advertisements
प्रश्न
Differentiate the following w.r.t. x: `x^(tan^(-1)x`
Advertisements
उत्तर
Let y = `x^(tan^(-1)x`
Then log y = `log (x^(tan^(-1)x)) = (tan^-1 x)(logx)`
Differentiating both sides w.r.t. x, we get
`(1)/y.(dy)/(dx) = d/(dx)[(tan^-1 x)(logx)]`
= `(tan^-1 x).d/(dx)(logx) + (logx).d/(dx)(tan^-1 x)`
= `(tan^-1 x) xx 1/x + (logx) xx 1/(1 + x^2)`
∴ `(dy)/(dx) = y[(tan^-1 x)/x + logx/(1 + x^2)]`
= `x^(tan^-1 x)[tan^-1 x/x + logx/(1 + x^2)]`
संबंधित प्रश्न
Differentiate the following w.r.t.x: `(8)/(3root(3)((2x^2 - 7x - 5)^11`
Differentiate the following w.r.t.x: cos(x2 + a2)
Differentiate the following w.r.t.x: `log[tan(x/2)]`
Differentiate the following w.r.t.x: `5^(sin^3x + 3)`
Differentiate the following w.r.t.x: log[cos(x3 – 5)]
Differentiate the following w.r.t.x: cos2[log(x2 + 7)]
Differentiate the following w.r.t.x: [log {log(logx)}]2
Differentiate the following w.r.t.x:
sin2x2 – cos2x2
Differentiate the following w.r.t.x:
(x2 + 4x + 1)3 + (x3− 5x − 2)4
Differentiate the following w.r.t.x: (1 + sin2 x)2 (1 + cos2 x)3
Differentiate the following w.r.t.x: `(1 + sinx°)/(1 - sinx°)`
Differentiate the following w.r.t. x : cosec–1 (e–x)
Differentiate the following w. r. t. x.
cos–1(1 – x2)
Differentiate the following w.r.t. x : `sin^-1(x^(3/2))`
Differentiate the following w.r.t. x :
`cos^-1(sqrt(1 - cos(x^2))/2)`
Differentiate the following w.r.t. x :
`cot^-1[(sqrt(1 + sin ((4x)/3)) + sqrt(1 - sin ((4x)/3)))/(sqrt(1 + sin ((4x)/3)) - sqrt(1 - sin ((4x)/3)))]`
Differentiate the following w.r.t. x : `sin^-1((1 - x^2)/(1 + x^2))`
Differentiate the following w.r.t. x:
`sin^-1 ((1 - 25x^2)/(1 + 25x^2))`
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 : `tan^-1((a + btanx)/(b - atanx))`
Differentiate the following w.r.t. x : `cot^-1((4 - x - 2x^2)/(3x + 2))`
Differentiate the following w.r.t. x :
`(x + 1)^2/((x + 2)^3(x + 3)^4`
Differentiate the following w.r.t. x :
etanx + (logx)tanx
Show that `"dy"/"dx" = y/x` in the following, where a and p are constants : x7.y5 = (x + y)12
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 : `tan^-1((3x^2 - 4y^2)/(3x^2 + 4y^2))` = a2
If y is a function of x and log (x + y) = 2xy, then the value of y'(0) = ______.
Differentiate y = `sqrt(x^2 + 5)` w.r. to x
Differentiate y = etanx w.r. to x
If y = `"e"^(1 + logx)` then find `("d"y)/("d"x)`
If y = `tan^-1[sqrt((1 + cos x)/(1 - cos x))]`, find `("d"y)/("d"x)`
Differentiate `tan^-1((8x)/(1 - 15x^2))` w.r. to x
If the function f(x) = `(log (1 + "ax") - log (1 - "bx))/x, x ≠ 0` is continuous at x = 0 then, f(0) = _____.
If `t = v^2/3`, then `(-v/2 (df)/dt)` is equal to, (where f is acceleration) ______
Derivative of (tanx)4 is ______
If x2 + y2 - 2axy = 0, then `dy/dx` equals ______
Let f(x) = `(1 - tan x)/(4x - pi), x ne pi/4, x ∈ [0, pi/2]`. If f(x) is continuous in `[0, pi/2]`, then f`(pi/4)` is ______.
The volume of a spherical balloon is increasing at the rate of 10 cubic centimetre per minute. The rate of change of the surface of the balloon at the instant when its radius is 4 centimetres, is ______
If x = p sin θ, y = q cos θ, then `dy/dx` = ______
If x = eθ, (sin θ – cos θ), y = eθ (sin θ + cos θ) then `dy/dx` at θ = `π/4` is ______.
Differentiate `tan^-1 (sqrt((3 - x)/(3 + x)))` w.r.t. x.
Diffierentiate: `tan^-1((a + b cos x)/(b - a cos x))` w.r.t.x.
If y = log (sec x + tan x), find `dy/dx`.
`lim_(x → 0) (sqrt(1 + x + x^2) − 1)/x` = ______.
