Advertisements
Advertisements
Question
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
Advertisements
Solution
`log((x^20 - y^20)/(x^20 + y^20))` = 20
∴ `(x^20 - y^20)/(x^20 + y^20)` = e20 = k ...(Say)
∴ x20 – y20 = kx20 + ky20
∴ (1 + k)y20 = kx20 + ky20
∴ `y^20/x^20 = (1 - k)/(1 + k)`
∴ `y/x = ((1 - k)/(1 + k))^(1/20)`, a constant
Differentiating both sides w.r.t. x, we get
`"d"/"dx"(y/x)` = 0
∴ `(x"dy"/"dx" - y."d"/"dx"(x))/(x^2)` = 0
∴ `x"dy"/"dx" - y xx 1` = 0
∴ `x"dy"/"dx"` = y
∴ `"dy"/"dx" = y/x`.
APPEARS IN
RELATED QUESTIONS
Differentiate the following w.r.t.x:
`(sqrt(3x - 5) - 1/sqrt(3x - 5))^5`
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: `sinsqrt(sinsqrt(x)`
Differentiate the following w.r.t.x: `log[sec (e^(x^2))]`
Differentiate the following w.r.t.x: `log_(e^2) (log x)`
Differentiate the following w.r.t.x: [log {log(logx)}]2
Differentiate the following w.r.t.x: log[tan3x.sin4x.(x2 + 7)7]
Differentiate the following w.r.t.x: `log(sqrt((1 - sinx)/(1 + sinx)))`
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:
y = (25)log5(secx) − (16)log4(tanx)
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 : `"cosec"^-1[1/cos(5^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 : `"cosec"^-1((1)/(4cos^3 2x - 3cos2x))`
Differentiate the following w.r.t. x : `tan^-1(sqrt((1 + cosx)/(1 - cosx)))`
Differentiate the following w.r.t. x : `sin^-1((4sinx + 5cosx)/sqrt(41))`
Differentiate the following w.r.t. x : `"cosec"^-1[(10)/(6sin(2^x) - 8cos(2^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 : `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 : `tan^-1((a + btanx)/(b - atanx))`
Differentiate the following w.r.t. x : `cot^-1((a^2 - 6x^2)/(5ax))`
Differentiate the following w.r.t. x :
`tan^-1((5 -x)/(6x^2 - 5x - 3))`
Differentiate the following w.r.t. x : `cot^-1((4 - x - 2x^2)/(3x + 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: (sin xx)
Differentiate the following w.r.t. x :
etanx + (logx)tanx
Differentiate the following w.r.t. x :
(sin x)tanx + (cos x)cotx
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
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)`
Differentiate sin2 (sin−1(x2)) w.r. to x
Differentiate `cot^-1((cos x)/(1 + sinx))` 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 f(x) = 3x - 2 and g(x) = x2, then (fog)(x) = ________.
Derivative of (tanx)4 is ______
If y = `(3x^2 - 4x + 7.5)^4, "then" dy/dx` is ______
The differential equation of the family of curves y = `"ae"^(2(x + "b"))` is ______.
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 ______
Solve `x + y (dy)/(dx) = sec(x^2 + y^2)`
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` = ______.
