Advertisements
Advertisements
प्रश्न
If y = log (log 2x), show that xy2 + y1 (1 + xy1) = 0.
उत्तर
y = log (log 2x)
∴ `"dy"/"dx" = "d"/"dx"[log(log2x)]`
= `(1)/"log2x"."d"/"dx"(log2x)`
= `(1)/"log2x" xx (1)/(2x)."d"/"dx"(2x)`
= `(1)/"log2x" xx (1)/(2x) xx 2`
∴ `"dy"/"dx" = (1)/(xlog2x)`
∴ `(log2x)."dy"/"dx" = (1)/x` ...(1)
Differentiating both sides w.r.t. x, we get
`(log2x)."d"/"dx"(dx/dy) + "dy"/"dx"."d"/"dx"(log2x) = "d"/"dx"(1/x)`
∴ `(log2x).(d^2y)/(dx^2) + "dy"/"dx".(1)/(2x)."d"/"dx"(2x) = -(1)/x^2`
∴ `(log2x).(d^2y)/(dx^2) + "dy"/"dx".(1)/(2x) xx 2 = -(1)/x^2`
∴ `(log2x).(d^2y)/(dx^2) + (1)/x."dy"/"dx" = (1)/x.(1)/x`
∴ `(log2x).(d^2y)/(dx^2) + [(log2x)."dy"/"dx"]"dy"/"dx" = -(1)/x[(log2x)."dy"/"dx"]` ...[By (1)]
Dividing throughout by log 2x, we get
`(d^2y)/(dx^2) + (dy/dx)^2 = -(1)/x"dy"/"dx"`
∴ `x(d^2y)/(dx^2) + x(dy/dx)^2 = -"dy"/"dx"`
∴ `x(d^2y)/(dx^2) + "dy"/"dx" + x(dy/dx)^2` = 0
∴ `x(d^2y)/(dx^2) + "dy"/"dx" (1 + xdy/dx)` = 0
∴ xy2 + y1 (1 + xy1) = 0.
APPEARS IN
संबंधित प्रश्न
if xx+xy+yx=ab, then find `dy/dx`.
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 cos x)^x + (x sin x)^(1/x)`
Find `dy/dx`for the function given in the question:
xy + yx = 1
Differentiate w.r.t. x the function:
xx + xa + ax + aa, for some fixed a > 0 and x > 0
If `y = sin^-1 x + cos^-1 x , "find" dy/dx`
Find `(dy)/(dx) , if y = sin ^(-1) [2^(x +1 )/(1+4^x)]`
Evaluate
`int 1/(16 - 9x^2) dx`
Find `dy/dx` if y = xx + 5x
If `(sin "x")^"y" = "x" + "y", "find" (d"y")/(d"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 = `x^(x^(x^(.^(.^.∞))`, then show that `"dy"/"dx" = y^2/(x(1 - logy).`.
If x = `asqrt(secθ - tanθ), y = asqrt(secθ + tanθ), "then show that" "dy"/"dx" = -y/x`.
If x = a cos3t, y = a sin3t, show that `"dy"/"dx" = -(y/x)^(1/3)`.
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.
Find the nth derivative of the following : log (ax + b)
If f(x) = logx (log x) then f'(e) is ______
If y = `log[4^(2x)((x^2 + 5)/sqrt(2x^3 - 4))^(3/2)]`, find `("d"y)/("d"x)`
The rate at which the metal cools in moving air is proportional to the difference of temperatures between the metal and air. If the air temperature is 290 K and the metal temperature drops from 370 K to 330 K in 1 O min, then the time required to drop the temperature upto 295 K.
lf y = `2^(x^(2^(x^(...∞))))`, then x(1 - y logx logy)`dy/dx` = ______
If y = tan-1 `((1 - cos 3x)/(sin 3x))`, then `"dy"/"dx"` = ______.
`"d"/"dx" [(cos x)^(log x)]` = ______.
Derivative of `log_6`x with respect 6x to is ______
`lim_("x" -> 0)(1 - "cos x")/"x"^2` is equal to ____________.
`lim_("x" -> -2) sqrt ("x"^2 + 5 - 3)/("x" + 2)` is equal to ____________.
If y `= "e"^(3"x" + 7), "then the value" |("dy")/("dx")|_("x" = 0)` is ____________.
If `f(x) = log [e^x ((3 - x)/(3 + x))^(1/3)]`, then `f^'(1)` is equal to
Given f(x) = `log((1 + x)/(1 - x))` and g(x) = `(3x + x^3)/(1 + 3x^2)`, then fog(x) equals
If `log_10 ((x^3 - y^3)/(x^3 + y^3))` = 2 then `dy/dx` = ______.
If `log_10 ((x^2 - y^2)/(x^2 + y^2))` = 2, then `dy/dx` is equal to ______.
Find the derivative of `y = log x + 1/x` with respect to x.
