Advertisements
Advertisements
प्रश्न
If y = `log(x + sqrt(x^2 + a^2))^m`, show that `(x^2 + a^2)(d^2y)/(dx^2) + x "d"/"dx"` = 0.
Advertisements
उत्तर
y = `log(x + sqrt(x^2 + a^2))^m`
= `mlog(x + sqrt(x^2 + a^2))`
∴ `"dy"/"dx" = m"d"/"dx"[log(x + sqrt(x^2 + a^2))]`
= `m xx (1)/(x + sqrt(x^2 + a^2))."d"/"dx"(x + sqrt(x^2 + a^2))`
= `m/(x + sqrt(x^2 + a^2)) xx [1 + (1)/(2sqrt(x^2 + a^2))."d"/"dx"(x^2 + a^2)]`
= `m/(x + sqrt(x^2 + a^2)) xx [1 + (1)/(2sqrt(x^2 + a^2)).(2x + 0)]`
= `m/(x + sqrt(x^2 + a^2)) xx (sqrt(x^2 + a^2) + x)/(sqrt(x^2 + a^2)`
∴ `"dy"/"dx" = m/sqrt(x^2 + a^2)`
∴ `sqrt(x^2 + a^2)"dy"/"dx"` = m
∴ `(x^2 + a^2)(dy/dx)^2` = m2
Differentiating both sides w.r.t. x, we get
`(x^2 + a^2)."d"/"dx"(dy/dx)^2 + (dy/dx)^2."d"/"dx"(x^2 + a^2) = "d"/"dx"(m^2)`
∴ `(x^2 + a^2) xx 2"dy"/"dx"."d"/"dx"(dy/dx) + (dy/dx)^2 xx (2x + 0)` = 0
∴ `(x^2 + a^2) . 2"dy"/"dx"(d^2y)/(dx^2) + 2x (dy/dx)^2` = 0
Cancelling `2"dy"/"dx"` throughtout, we get
`(x^2 + a^2)(d^2y)/(dx^2) + x"dy"/"dx"` = 0.
संबंधित प्रश्न
If `y=log[x+sqrt(x^2+a^2)]` show that `(x^2+a^2)(d^2y)/(dx^2)+xdy/dx=0`
Differentiate the function with respect to x.
cos x . cos 2x . cos 3x
Differentiate the function with respect to x.
`(sin x)^x + sin^(-1) sqrtx`
Find `bb(dy/dx)` for the given function:
xy = `e^((x - y))`
Find the derivative of the function given by f(x) = (1 + x) (1 + x2) (1 + x4) (1 + x8) and hence find f′(1).
If cos y = x cos (a + y), with cos a ≠ ± 1, prove that `dy/dx = cos^2(a+y)/(sin a)`.
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 `x^m y^n = (x + y)^(m + n)`, prove that `(d^2y)/(dx^2)= 0`
If ey ( x +1) = 1, then show that `(d^2 y)/(dx^2) = ((dy)/(dx))^2 .`
Find `(dy)/(dx) , if y = sin ^(-1) [2^(x +1 )/(1+4^x)]`
Find `dy/dx` if y = xx + 5x
xy = ex-y, then show that `"dy"/"dx" = ("log x")/("1 + log x")^2`
Differentiate : log (1 + x2) w.r.t. cot-1 x.
Solve the following differential equation: (3xy + y2) dx + (x2 + xy) dy = 0
If y = (log x)x + xlog x, find `"dy"/"dx".`
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 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 = 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 : x3.logx
Find the nth derivative of the following : log (2x + 3)
If y = A cos (log x) + B sin (log x), show that x2y2 + xy1 + y = 0.
If y = `25^(log_5sin_x) + 16^(log_4cos_x)` then `("d"y)/("d"x)` = ______.
If y = `log[4^(2x)((x^2 + 5)/sqrt(2x^3 - 4))^(3/2)]`, 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 x7 . y5 = (x + y)12, show that `("d"y)/("d"x) = y/x`
If y = `(sin x)^sin x` , then `"dy"/"dx"` = ?
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.
Derivative of loge2 (logx) with respect to x is _______.
If y = `{f(x)}^{phi(x)}`, then `dy/dx` is ______
`2^(cos^(2_x)`
`8^x/x^8`
If y = `log ((1 - x^2)/(1 + x^2))`, then `"dy"/"dx"` is equal to ______.
If `"f" ("x") = sqrt (1 + "cos"^2 ("x"^2)), "then the value of f'" (sqrtpi/2)` is ____________.
If y `= "e"^(3"x" + 7), "then the value" |("dy")/("dx")|_("x" = 0)` is ____________.
If `log_10 ((x^3 - y^3)/(x^3 + y^3))` = 2 then `dy/dx` = ______.
Derivative of log (sec θ + tan θ) with respect to sec θ at θ = `π/4` is ______.
Find `dy/dx`, if y = (sin x)tan x – xlog x.
If y = `9^(log_3x)`, find `dy/dx`.
Find `dy/dx`, if y = (log x)x.
