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:
(x3 – 2x – 1)5
Differentiate the following w.r.t.x: `(8)/(3root(3)((2x^2 - 7x - 5)^11`
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 {log(logx)}]2
Differentiate the following w.r.t.x:
`sqrt(cosx) + sqrt(cossqrt(x)`
Differentiate the following w.r.t.x: `(1 + sinx°)/(1 - sinx°)`
Differentiate the following w.r.t.x: log[tan3x.sin4x.(x2 + 7)7]
Differentiate the following w.r.t.x:
`log(sqrt((1 - cos3x)/(1 + cos3x)))`
Differentiate the following w.r.t.x: `log[4^(2x)((x^2 + 5)/(sqrt(2x^3 - 4)))^(3/2)]`
Differentiate the following w.r.t. x : cot–1(4x)
Differentiate the following w.r.t. x : `tan^-1(sqrt(x))`
Differentiate the following w. r. t. x.
cos–1(1 – x2)
Differentiate the following w.r.t. x : `"cosec"^-1((1)/(4cos^3 2x - 3cos2x))`
Differentiate the following w.r.t. x : `sin^-1((4sinx + 5cosx)/sqrt(41))`
Differentiate the following w.r.t. x : `sin^-1((cossqrt(x) + sinsqrt(x))/sqrt(2))`
Differentiate the following w.r.t. x :
`sin^-1(4^(x + 1/2)/(1 + 2^(4x)))`
Differentiate the following w.r.t. x :
`sin^(−1) ((1 − x^3)/(1 + x^3))`
Differentiate the following w.r.t. x : `tan^-1((2sqrt(x))/(1 + 3x))`
Differentiate the following w.r.t. x : `tan^-1((2^x)/(1 + 2^(2x + 1)))`
Differentiate the following w.r.t. x : `root(3)((4x - 1)/((2x + 3)(5 - 2x)^2)`
Differentiate the following w.r.t. x : `(x^2 + 3)^(3/2).sin^3 2x.2^(x^2)`
Differentiate the following w.r.t. x : `((x^2 + 2x + 2)^(3/2))/((sqrt(x) + 3)^3(cosx)^x`
Differentiate the following w.r.t. x :
etanx + (logx)tanx
Differentiate the following w.r.t. x : `10^(x^(x)) + x^(x(10)) + x^(10x)`
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 : `sec((x^5 + y^5)/(x^5 - y^5))` = a2
Show that `"dy"/"dx" = y/x` in the following, where a and p are constants: `log((x^20 - y^20)/(x^20 + y^20))` = 20
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 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
If f(x) is odd and differentiable, then f′(x) is
Differentiate `sin^-1((2cosx + 3sinx)/sqrt(13))` w.r. to x
Differentiate `tan^-1((8x)/(1 - 15x^2))` w.r. to x
If y = `sqrt(cos x + sqrt(cos x + sqrt(cos x + ...... ∞)`, show that `("d"y)/("d"x) = (sin x)/(1 - 2y)`
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) ______
y = {x(x - 3)}2 increases for all values of x lying in the interval.
A particle moves so that x = 2 + 27t - t3. The direction of motion reverses after moving a distance of ______ units.
If y = cosec x0, then `"dy"/"dx"` = ______.
If x = p sin θ, y = q cos θ, then `dy/dx` = ______
Solve `x + y (dy)/(dx) = sec(x^2 + y^2)`
`lim_(x → 0) (sqrt(1 + x + x^2) − 1)/x` = ______.
