Advertisements
Advertisements
Question
If `y=log[x+sqrt(x^2+a^2)]` show that `(x^2+a^2)(d^2y)/(dx^2)+xdy/dx=0`
Advertisements
Solution
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
RELATED QUESTIONS
Differentiate the following function with respect to x: `(log x)^x+x^(logx)`
Differentiate the function with respect to x.
(log x)cos x
Differentiate the function with respect to x.
xx − 2sin x
Find `bb(dy/dx)` for the given function:
(cos x)y = (cos y)x
Find `bb(dy/dx)` for the given function:
xy = `e^((x - y))`
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?
Find `dy/dx` if y = xx + 5x
If y = (log x)x + xlog x, find `"dy"/"dx".`
`"If" y = sqrt(logx + sqrt(log x + sqrt(log x + ... ∞))), "then show that" dy/dx = (1)/(x(2y - 1).`
If x = a cos3t, y = a sin3t, show that `"dy"/"dx" = -(y/x)^(1/3)`.
If x = log(1 + t2), y = t – tan–1t,show that `"dy"/"dx" = sqrt(e^x - 1)/(2)`.
Find the second order derivatives of the following : x3.logx
If y = log (log 2x), show that xy2 + y1 (1 + xy1) = 0.
Find the nth derivative of the following : log (2x + 3)
If f(x) = logx (log x) then f'(e) is ______
If y = `log[sqrt((1 - cos((3x)/2))/(1 +cos((3x)/2)))]`, find `("d"y)/("d"x)`
If `("f"(x))/(log (sec x)) "dx"` = log(log sec x) + c, then f(x) = ______.
`log (x + sqrt(x^2 + "a"))`
If xm . yn = (x + y)m+n, prove that `"dy"/"dx" = y/x`
`lim_("x" -> 0)(1 - "cos x")/"x"^2` is equal to ____________.
If `f(x) = log [e^x ((3 - x)/(3 + x))^(1/3)]`, then `f^'(1)` is equal to
If `log_10 ((x^2 - y^2)/(x^2 + y^2))` = 2, then `dy/dx` is equal to ______.
If y = `log(x + sqrt(x^2 + 4))`, show that `dy/dx = 1/sqrt(x^2 + 4)`
The derivative of log x with respect to `1/x` is ______.
Find the derivative of `y = log x + 1/x` with respect to x.
If xy = yx, then find `dy/dx`
What condition must be ensured for an expression inside logarithm?
