Advertisements
Advertisements
प्रश्न
Differentiate the following w.r.t. x : `((x^2 + 2x + 2)^(3/2))/((sqrt(x) + 3)^3(cosx)^x`
Advertisements
उत्तर
Let y = `((x^2 + 2x + 2)^(3/2))/((sqrt(x) + 3)^3(cosx)^x`
Then log y = `log[((x^2 + 2x + 2)^(3/2))/((sqrt(x) + 3)^3(cosx)^x)]`
= `log(x^2 + 2x + 2)^(3/2) - log(sqrt(x) + 3)^3(cosx)^x`
= `(3)/(2)log(x^2 + 2x + 2) - 3log(sqrt(x) + 3) - xlog(cosx)`
Differentiating both sides w.r.t. x, we get
`(1)/y."dy"/"dx" = (3)/(2)"d"/"dx"[log(x^2 + 2x + 2)] -3"d"/"dx"[log(sqrt(x) + 3)] - "d"/"dx"[xlog(cosx)]`
= `(3)/(2) xx (1)/(x^2 + 2x + 2)."d"/"dx"(x^2 + 2x + 2) -3 xx (1)/(sqrt(x) + 3)."d"/"dx"(sqrt(x) + 3) - {x"d"/"dx"[log(cosx)] + log(cosx)."d"/"dx"(x)}`
= `(3)/(2(x^2 + 2x + 2)) xx (2x + 2 xx 1 + 0) - (3)/(sqrt(x) + 3) xx (1/(2sqrt(x)) + 0) - {x xx (1)/cosx."d"/"dx"(cosx) + log(cosx) xx 1}`
= `(3(2x + 2))/(2(x^2 + 2x + 2)) - (3)/(2sqrt(x)(sqrt(x) + 3)) - {x xx (1)/cosx.(-sinx) + log(cosx)}`
∴ `"dy"/"dx" = y[(3(x + 1))/(x^2 + 2x + 2) - (3)/(2sqrt(x)(sqrt(x) + 3)) + xtanx - log(cosx)]`
= `((x^2 + 2x + 2)^(3/2))/((sqrt(x) + 3)^3(cosx)^x)[(3(x + 1))/(x^2 + 2x + 2) - (3)/(2sqrt(x)(sqrt(x) + 3)) + xtanx - log(cosx)]`.
APPEARS IN
संबंधित प्रश्न
Differentiate the following w.r.t.x:
`sqrt(e^((3x + 2) + 5)`
Differentiate the following w.r.t.x: cos2[log(x2 + 7)]
Differentiate the following w.r.t.x: sec[tan (x4 + 4)]
Differentiate the following w.r.t.x: `sinsqrt(sinsqrt(x)`
Differentiate the following w.r.t.x: `x/(sqrt(7 - 3x)`
Differentiate the following w.r.t.x:
log (sec 3x+ tan 3x)
Differentiate the following w.r.t.x: `log(sqrt((1 - sinx)/(1 + sinx)))`
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: `log[(ex^2(5 - 4x)^(3/2))/root(3)(7 - 6x)]`
Differentiate the following w.r.t.x:
`log[a^(cosx)/((x^2 - 3)^3 logx)]`
Differentiate the following w.r.t. x :
cos3[cos–1(x3)]
Differentiate the following w.r.t. x : `cot^-1[cot(e^(x^2))]`
Differentiate the following w.r.t. x : `cos^-1(sqrt((1 + cosx)/2))`
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 : `cos^-1((3cos3x - 4sin3x)/5)`
Differentiate the following w.r.t. x : cos–1(3x – 4x3)
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 :
`tan^(−1)[(2^(x + 2))/(1 − 3(4^x))]`
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 : `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^(tan^(-1)x`
Differentiate the following w.r.t. x:
`x^(x^x) + e^(x^x)`
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
Show that `"dy"/"dx" = y/x` in the following, where a and p are constants : `cos^-1((7x^4 + 5y^4)/(7x^4 - 5y^4)) = tan^-1a`
Differentiate y = `sqrt(x^2 + 5)` w.r. to x
Differentiate y = etanx w.r. to x
If f(x) is odd and differentiable, then f′(x) is
If y = `"e"^(1 + logx)` then find `("d"y)/("d"x)`
Differentiate sin2 (sin−1(x2)) 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 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) = ________.
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 ______
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 ______
Solve `x + y (dy)/(dx) = sec(x^2 + y^2)`
Find `(dy)/(dx)`, if x3 + x2y + xy2 + y3 = 81
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`
If y = log (sec x + tan x), find `dy/dx`.
