Advertisements
Advertisements
प्रश्न
If y = `x^(x^(x^(.^(.^.∞))`, then show that `"dy"/"dx" = y^2/(x(1 - logy).`.
Advertisements
उत्तर
y = `x^(x^(x^(.^(.^.∞))`
∴ log y = `log(x^(x^(x^(.^(.^.∞)))))`
= `x^(x^(x^(.^(.^.∞)))).logx`
∴ log y = y log x ...(1)
Differentiating both sides w.r.t. x, we get
`(1)/y.dy/dx = y.d/dx(logx) + (logx)dy/dx`
∴ `(1)/ydy/dx = y xx (1)/x + (logx)dy/dx`
∴ `(1/y - logx)dy/dx = y/x`
∴ `((1 - ylogx)/(y))dy/dx"= y/x`
∴ `dy/dx = y^2/(x(1 - ylogx)`
∴ `dy/dx = y^2/(x(1 - logy)`. ...[By (1)]
APPEARS IN
संबंधित प्रश्न
Differentiate the following function with respect to x: `(log x)^x+x^(logx)`
If `y=log[x+sqrt(x^2+a^2)]` show that `(x^2+a^2)(d^2y)/(dx^2)+xdy/dx=0`
if xx+xy+yx=ab, then find `dy/dx`.
Differentiate the function with respect to x.
(log x)x + xlog x
Differentiate the function with respect to x.
`(sin x)^x + sin^(-1) sqrtx`
Differentiate the function with respect to x.
xsin x + (sin x)cos x
Differentiate the function with respect to x.
`(x cos x)^x + (x sin x)^(1/x)`
Find `bb(dy/dx)` for the given function:
(cos x)y = (cos y)x
If cos y = x cos (a + y), with cos a ≠ ± 1, prove that `dy/dx = cos^2(a+y)/(sin a)`.
If `y = sin^-1 x + cos^-1 x , "find" dy/dx`
Evaluate
`int 1/(16 - 9x^2) dx`
Find `dy/dx` if y = xx + 5x
Find `(d^2y)/(dx^2)` , if y = log x
Find `"dy"/"dx"` , if `"y" = "x"^("e"^"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 log (x + y) = log(xy) + p, where p is a constant, then prove that `"dy"/"dx" = (-y^2)/(x^2)`.
If `log_10((x^3 - y^3)/(x^3 + y^3))` = 2, show that `dy/dx = -(99x^2)/(101y^2)`.
If `log_5((x^4 + y^4)/(x^4 - y^4)) = 2, "show that""dy"/"dx" = (12x^3)/(13y^3)`.
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 = esin3t, y = ecos3t, then show that `dy/dx = -(ylogx)/(xlogy)`.
If x = 2cos4(t + 3), y = 3sin4(t + 3), show that `"dy"/"dx" = -sqrt((3y)/(2x)`.
Find the second order derivatives of the following : log(logx)
If y = `log(x + sqrt(x^2 + a^2))^m`, show that `(x^2 + a^2)(d^2y)/(dx^2) + x "d"/"dx"` = 0.
If y = log (log 2x), show that xy2 + y1 (1 + xy1) = 0.
Find the nth derivative of the following: log (ax + b)
If y = `25^(log_5sin_x) + 16^(log_4cos_x)` then `("d"y)/("d"x)` = ______.
If y = `log[sqrt((1 - cos((3x)/2))/(1 +cos((3x)/2)))]`, find `("d"y)/("d"x)`
If log5 `((x^4 + "y"^4)/(x^4 - "y"^4))` = 2, show that `("dy")/("d"x) = (12x^3)/(13"y"^2)`
If y = 5x. x5. xx. 55 , find `("d"y)/("d"x)`
Derivative of loge2 (logx) with respect to x is _______.
If xy = ex-y, then `"dy"/"dx"` at x = 1 is ______.
`"d"/"dx" [(cos x)^(log x)]` = ______.
`log (x + sqrt(x^2 + "a"))`
If `"f" ("x") = sqrt (1 + "cos"^2 ("x"^2)), "then the value of f'" (sqrtpi/2)` is ____________.
If `"y" = "e"^(1/2log (1 + "tan"^2"x")), "then" "dy"/"dx"` is equal to ____________.
If `f(x) = log [e^x ((3 - x)/(3 + x))^(1/3)]`, then `f^'(1)` is equal to
If y = `log(x + sqrt(x^2 + 4))`, show that `dy/dx = 1/sqrt(x^2 + 4)`
If y = `9^(log_3x)`, find `dy/dx`.
If \[y=x^x+x^{\frac{1}{x}}\] then \[\frac{\mathrm{d}y}{\mathrm{d}x}\] is equal to
