Advertisements
Advertisements
प्रश्न
If x = esin3t, y = ecos3t, then show that `dy/dx = -(ylogx)/(xlogy)`.
उत्तर
x = esin3t, y = ecos3t
∴ log x = logesin3t, logy = logecos3t
∴ log x = (sin 3t)(log e), log y = (cos 3t)(log e)
∴ log x = sin 3t, log y = cos 3t ...(1) ... [∵ log e = 1]
Differentiating both sides w.r.t. t, we get
`(1)/x.dx/dt = d/dt(sin3t) = cos3t.d/dt(3t)`
= cos 3t x 3
= 3 cos 3t
and
`(1)/y.dy/dt = d/dt(cos 3t) = -sin3t.d/dx(3t)`
= – sin 3t x 3
= – 3 sin 3t
∴ `dx/dt = 3x cos 3t and dy/dt"= -3y sin 3t`
∴ `dy/dx = ((dy/dt))/((dx/dt)`
= `(-3y sin 3t)/(3x cos 3t)`
= `(-y sin 3t)/(x cos 3t)`
= `(-y log x)/(x log y)`. ...[By (1)]
APPEARS IN
संबंधित प्रश्न
Differentiate the following function with respect to x: `(log x)^x+x^(logx)`
Differentiate the function with respect to x.
cos x . cos 2x . cos 3x
Differentiate the function with respect to x.
`sqrt(((x-1)(x-2))/((x-3)(x-4)(x-5)))`
Differentiate the function with respect to x.
`x^x - 2^(sin x)`
Differentiate the function with respect to x.
(x + 3)2 . (x + 4)3 . (x + 5)4
Differentiate the function with respect to x.
`(x + 1/x)^x + x^((1+1/x))`
Find `dy/dx` for the function given in the question:
yx = xy
Find the derivative of the function given by f (x) = (1 + x) (1 + x2) (1 + x4) (1 + x8) and hence find f ′(1).
If x = a (cos t + t sin t) and y = a (sin t – t cos t), find `(d^2y)/dx^2`
If `y = e^(acos^(-1)x)`, -1 <= x <= 1 show that `(1- x^2) (d^2y)/(dx^2) -x dy/dx - a^2y = 0`
If ey ( x +1) = 1, then show that `(d^2 y)/(dx^2) = ((dy)/(dx))^2 .`
Find `"dy"/"dx"` , if `"y" = "x"^("e"^"x")`
Differentiate : log (1 + x2) w.r.t. cot-1 x.
Find `"dy"/"dx"` if y = xx + 5x
If `"x"^(5/3) . "y"^(2/3) = ("x + y")^(7/3)` , the show that `"dy"/"dx" = "y"/"x"`
If y = (log x)x + xlog x, find `"dy"/"dx".`
If `log_10((x^3 - y^3)/(x^3 + y^3)) = 2, "show that" "dy"/"dx" = -(99x^2)/(101y^2)`
If xy = ex–y, then show that `"dy"/"dx" = logx/(1 + logx)^2`.
`"If" y = sqrt(logx + sqrt(log x + sqrt(log x + ... ∞))), "then show that" dy/dx = (1)/(x(2y - 1).`
If x = sin–1(et), y = `sqrt(1 - e^(2t)), "show that" sin x + dy/dx` = 0
Differentiate 3x w.r.t. logx3.
Find the second order derivatives of the following : x3.logx
Find the second order derivatives of the following : log(logx)
If y = log (log 2x), show that xy2 + y1 (1 + xy1) = 0.
Find the nth derivative of the following : log (ax + b)
If f(x) = logx (log x) then f'(e) is ______
If y = `(sin x)^sin x` , then `"dy"/"dx"` = ?
Derivative of loge2 (logx) with respect to x is _______.
lf y = `2^(x^(2^(x^(...∞))))`, then x(1 - y logx logy)`dy/dx` = ______
`"d"/"dx" [(cos x)^(log x)]` = ______.
If y = `("e"^"2x" sin x)/(x cos x), "then" "dy"/"dx" = ?`
Given f(x) = `log((1 + x)/(1 - x))` and g(x) = `(3x + x^3)/(1 + 3x^2)`, then fog(x) equals
If y = `x^(x^2)`, then `dy/dx` is equal to ______.
If `log_10 ((x^2 - y^2)/(x^2 + y^2))` = 2, then `dy/dx` is equal to ______.
If y = `log(x + sqrt(x^2 + 4))`, show that `dy/dx = 1/sqrt(x^2 + 4)`
Find `dy/dx`, if y = (log x)x.
Find the derivative of `y = log x + 1/x` with respect to x.
