Advertisements
Advertisements
प्रश्न
Differentiate the following w.r.t. x : `root(3)((4x - 1)/((2x + 3)(5 - 2x)^2)`
Advertisements
उत्तर
Let y = `root(3)((4x - 1)/((2x + 3)(5 - 2x)^2)`
Then log y = `log[(4x - 1)/((2x + 3)(5 - 2x)^2)]^(1/3)`
= `(1)/(3)log[(4x - 1)/((2x + 3)(5 - 2x)^2)]`
= `(1)/(3)[log(4x - 1) - log(2x + 3)(5 - 2x)^2]`
= `(1)/(3)log(4x - 1) - (1)/(3)log(2x + 3) - (2)/(3)log(5 - 2x)`
Differentiating both sides w.r.t. x, we get
`(1)/y."dy"/"dx" = (1)/(3)"d"/"dx"[log(4x - 1)] - (1)/(3)"d"/"dx"[log(2x + 3)] - (2)/(3)"d"/"dx"[log(5 - 2x)]`
= `(1)/(3) xx (1)/(4x - 1)."d"/"dx"(4x - 1) - (1)/(3) xx (1)/(2x + 3)."d"/"dx"(2x + 3) - (2)/(3) xx (1)/(5 - 2x)."d"/"dx"(5 - 2x)`
= `(1)/(3(4x - 1)). (4 xx 1 - 0) - (1)/(3(2x + 3)).(2 xx 1 + 0) - (2)/(3(5 - 2x)).(0 - 2 xx 1)`
∴ `"dy"/"dx" = y[(4)/(3(4x - 1)) - (2)/(3(2x + 3)) + (4)/(3(5 - 2x))]`
= `root(3)((4x - 1)/((2x + 3)(5 - 2x)^2))[(4)/(3(4x - 1)) - (2)/(3(2x + 3)) + (4)/(3(5 - 2x))]`.
APPEARS IN
संबंधित प्रश्न
Differentiate the following w.r.t.x:
`(2x^(3/2) - 3x^(4/3) - 5)^(5/2)`
Differentiate the following w.r.t.x: `(8)/(3root(3)((2x^2 - 7x - 5)^11`
Differentiate the following w.r.t.x:
`sqrt(e^((3x + 2) + 5)`
Differentiate the following w.r.t.x: `sqrt(tansqrt(x)`
Differentiate the following w.r.t.x: cot3[log(x3)]
Differentiate the following w.r.t.x: cos2[log(x2 + 7)]
Differentiate the following w.r.t.x:
tan[cos(sinx)]
Differentiate the following w.r.t.x: `e^(log[(logx)^2 - logx^2]`
Differentiate the following w.r.t.x: [log {log(logx)}]2
Differentiate the following w.r.t.x: `(e^sqrt(x) + 1)/(e^sqrt(x) - 1)`
Differentiate the following w.r.t.x:
`log(sqrt((1 - cos3x)/(1 + cos3x)))`
Differentiate the following w.r.t. x : tan–1(log x)
Differentiate the following w.r.t. x : cot–1(x3)
Differentiate the following w.r.t. x : `tan^-1(sqrt(x))`
Differentiate the following w.r.t. x : `sin^4[sin^-1(sqrt(x))]`
Differentiate the following w.r.t. x : `"cosec"^-1[1/cos(5^x)]`
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((cossqrt(x) + sinsqrt(x))/sqrt(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((1 - x^2)/(1 + x^2))`
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(4^(x + 1/2)/(1 + 2^(4x)))`
Differentiate the following w.r.t. x : `cot^-1((a^2 - 6x^2)/(5ax))`
Differentiate the following w.r.t. x :
`(x + 1)^2/((x + 2)^3(x + 3)^4`
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: xe + xx + ex + ee.
Differentiate the following w.r.t. x :
etanx + (logx)tanx
Differentiate the following w.r.t. x :
(sin x)tanx + (cos x)cotx
Differentiate the following w.r.t. x : `10^(x^(x)) + x^(x(10)) + x^(10x)`
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 : `cos^-1((7x^4 + 5y^4)/(7x^4 - 5y^4)) = tan^-1a`
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) = ______.
If y = `tan^-1[sqrt((1 + cos x)/(1 - cos x))]`, find `("d"y)/("d"x)`
If y = `sin^-1[("a"cosx - "b"sinx)/sqrt("a"^2 + "b"^2)]`, then 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) = _____.
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 ______
If y = cosec x0, then `"dy"/"dx"` = ______.
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 `cos((x^2 - y^2)/(x^2 + y^2))` = log a, show that `dy/dx = y/x`
