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.
(x + 3)2 . (x + 4)3 . (x + 5)4
Differentiate the function with respect to x.
`(sin x)^x + sin^(-1) sqrtx`
Find `bb(dy/dx)` for the given function:
yx = xy
Differentiate (x2 – 5x + 8) (x3 + 7x + 9) in three ways mentioned below:
- By using the product rule.
- By expanding the product to obtain a single polynomial.
- By logarithmic differentiation.
Do they all give the same answer?
If u, v and w are functions of x, then show that `d/dx(u.v.w) = (du)/dx v.w + u. (dv)/dx.w + u.v. (dw)/dx` in two ways-first by repeated application of product rule, second by logarithmic differentiation.
Find `(dy)/(dx) , if y = sin ^(-1) [2^(x +1 )/(1+4^x)]`
Find `dy/dx` if y = xx + 5x
Find `(d^2y)/(dx^2)` , if y = log x
Solve the following differential equation: (3xy + y2) dx + (x2 + xy) dy = 0
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 y = `x^(x^(x^(.^(.^.∞))`, then show that `"dy"/"dx" = y^2/(x(1 - logy).`.
If ey = yx, then show that `"dy"/"dx" = (logy)^2/(log y - 1)`.
If x = a cos3t, y = a sin3t, show that `"dy"/"dx" = -(y/x)^(1/3)`.
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.
If y = `25^(log_5sin_x) + 16^(log_4cos_x)` then `("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`
`d/dx(x^{sinx})` = ______
`log (x + sqrt(x^2 + "a"))`
`log [log(logx^5)]`
`lim_("x" -> 0)(1 - "cos x")/"x"^2` is equal to ____________.
If y = `x^(x^2)`, then `dy/dx` is equal to ______.
If y = `9^(log_3x)`, find `dy/dx`.
If xy = yx, then find `dy/dx`
For which type of function is logarithmic differentiation especially useful?
For \[y=x^{\sin x}\], \[x>0\], what equation results after taking logarithm on both sides?
What is \[\frac{dy}{dx}\] for \[y=x^{\sin x}\], \[x>0\]?
