Advertisements
Advertisements
प्रश्न
Differentiate the following w.r.t. x :
`tan^(−1)[(2^(x + 2))/(1 − 3(4^x))]`
Advertisements
उत्तर
Let `y = tan^(−1)[(2^(x + 2))/(1 − 3(4^x))]`
`y = tan^(−1)[(2^2 . 2^x)/(1 − 3(4^x))]`
`y = tan^(−1)[(4.2^x)/(1 − 3(2^x)^2)]`
`y = tan^(−1)[(3 × 2^x + 1 × 2^x)/(1 − 3.2^x × 1.2^x)] ...[tan^(−1) x + tan^(−1) y = tan^(-1) ((x + y)/(1 - xy))]`
y = tan–1(3.2x) + tan–1(2x)
Differentiating w.r.t. x, we get
`dy/dx = d/dx [tan^-1 (3.2^x) + tan^-1(2^x)]`
`dy/dx = d/dx [tan^-1(3.2^x)] + d/dx [tan^-1 (2^x)]`
`dy/dx = (1)/(1 + (3.2^x)^2). d/dx (3.2^x) + (1)/(1 + (2^x)^2). d/dx (2^x)`
`dy/dx = (1)/(1 + 9(2^(2x))) × 3 × 2^xlog2 + (1)/(1 + 2^(2x)) × 2^xlog2`
`dy/dx = 2^xlog2[(3)/(1 + 9(2^(2x))) + (1)/(1 + 2^(2x))]`
APPEARS IN
संबंधित प्रश्न
Differentiate the following w.r.t.x:
`(sqrt(3x - 5) - 1/sqrt(3x - 5))^5`
Differentiate the following w.r.t.x: `log[tan(x/2)]`
Differentiate the following w.r.t.x: cot3[log(x3)]
Differentiate the following w.r.t.x: `e^(3sin^2x - 2cos^2x)`
Differentiate the following w.r.t.x: cos2[log(x2 + 7)]
Differentiate the following w.r.t.x:
sin2x2 – cos2x2
Differentiate the following w.r.t.x:
`(x^3 - 5)^5/(x^3 + 3)^3`
Differentiate the following w.r.t.x: (1 + sin2 x)2 (1 + cos2 x)3
Differentiate the following w.r.t.x:
log (sec 3x+ tan 3x)
Differentiate the following w.r.t.x:
`(e^(2x) - e^(-2x))/(e^(2x) + e^(-2x))`
Differentiate the following w.r.t.x: `(e^sqrt(x) + 1)/(e^sqrt(x) - 1)`
Differentiate the following w.r.t.x: log[tan3x.sin4x.(x2 + 7)7]
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 : `tan^-1(sqrt(x))`
Differentiate the following w.r.t. x : `tan^-1[(1 - tan(x/2))/(1 + tan(x/2))]`
Differentiate the following w.r.t. x : `tan^-1(sqrt((1 + cosx)/(1 - cosx)))`
Differentiate the following w.r.t. x : `cos^-1((sqrt(3)cosx - sinx)/(2))`
Differentiate the following w.r.t. x :
`cos^-1[(3cos(e^x) + 2sin(e^x))/sqrt(13)]`
Differentiate the following w.r.t. x : `tan^-1((2x)/(1 - x^2))`
Differentiate the following w.r.t. x : `sin^-1(2xsqrt(1 - x^2))`
Differentiate the following w.r.t. x :
`cos^-1 ((1 - 9^x))/((1 + 9^x)`
Differentiate the following w.r.t. x:
`sin^-1 ((1 - 25x^2)/(1 + 25x^2))`
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 : `tan^-1((2sqrt(x))/(1 + 3x))`
Differentiate the following w.r.t. x : (sin x)x
Show that `"dy"/"dx" = y/x` in the following, where a and p are constants : x7.y5 = (x + y)12
If y = sin−1 (2x), find `("d"y)/(""d"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) ______
A particle moves so that x = 2 + 27t - t3. The direction of motion reverses after moving a distance of ______ units.
If f(x) = `(3x + 1)/(5x - 4)` and t = `(5 + 3x)/(x - 4)`, then f(t) is ______
If x2 + y2 - 2axy = 0, then `dy/dx` equals ______
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 = eθ, (sin θ – cos θ), y = eθ (sin θ + cos θ) then `dy/dx` at θ = `π/4` 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 ______.
If `cos((x^2 - y^2)/(x^2 + y^2))` = log a, show that `dy/dx = y/x`
`lim_(x → 0) (sqrt(1 + x + x^2) − 1)/x` = ______.
