Advertisements
Advertisements
प्रश्न
If `y=log[x+sqrt(x^2+a^2)]` show that `(x^2+a^2)(d^2y)/(dx^2)+xdy/dx=0`
Advertisements
उत्तर
It is given that:
`y=log[x+sqrt(x^2+a^2)]`
Differentiating equation (1) with respect to x, we get
`dy/dx=(1+x/sqrt(x^2+a^2))/(x+sqrt(x^2+a^2))`
`dy/dx=1/sqrt(x^2+a^2)........(2)`
`xdy/dx=x/sqrt(x^2+a^2).........(3)`
Again differentiating equation (2) with respect to x, we get
`(d^2y)/(dx^2)=-x/(x^2+a^2)^(3/2)`
`(x^2+y^2)(d^2y)/(dx^2)=-x/sqrt(x^2+a^2)............(4)`
Adding equation (3) and (4), we get
`(x^2+y^2)(d^2y)/(dx^2)+xdy/dx=-x/sqrt(x^2+a^2)+x/sqrt(x^2+a^2)=0`
`(x^2+y^2)(d^2y)/(dx^2)+xdy/dx=0`
APPEARS IN
संबंधित प्रश्न
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 + 3)2 . (x + 4)3 . (x + 5)4
Differentiate the function with respect to x.
(log x)x + xlog x
Find `bb(dy/dx)` for the given function:
yx = xy
If x = a (cos t + t sin t) and y = a (sin t – t cos t), find `(d^2y)/dx^2`.
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)]`
Evaluate
`int 1/(16 - 9x^2) dx`
Find `(d^2y)/(dx^2)` , if y = log x
xy = ex-y, then show that `"dy"/"dx" = ("log x")/("1 + log x")^2`
If `(sin "x")^"y" = "x" + "y", "find" (d"y")/(d"x")`
If x = log(1 + t2), y = t – tan–1t,show that `"dy"/"dx" = sqrt(e^x - 1)/(2)`.
If x = sin–1(et), y = `sqrt(1 - e^(2t)), "show that" sin x + dy/dx` = 0
If x = `(2bt)/(1 + t^2), y = a((1 - t^2)/(1 + t^2)), "show that" "dx"/"dy" = -(b^2y)/(a^2x)`.
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 f(x) = logx (log x) then f'(e) is ______
If y = log [cos(x5)] then find `("d"y)/("d"x)`
If y = 5x. x5. xx. 55 , find `("d"y)/("d"x)`
If y = tan-1 `((1 - cos 3x)/(sin 3x))`, then `"dy"/"dx"` = ______.
If `("f"(x))/(log (sec x)) "dx"` = log(log sec x) + c, then f(x) = ______.
`2^(cos^(2_x)`
If y = `log ((1 - x^2)/(1 + x^2))`, then `"dy"/"dx"` is equal to ______.
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 y = `x^(x^2)`, then `dy/dx` is equal to ______.
Find `dy/dx`, if y = (sin x)tan x – xlog x.
If \[y=[u(x)]^{v(x)}\], which expression gives \[\frac{dy}{dx}\] in logarithmic differentiation?
After differentiating \[\log y=v(x)\cdot\log[u(x)]\], which equation is obtained?
