Advertisements
Advertisements
प्रश्न
Find the second order derivatives of the following : log(logx)
Advertisements
उत्तर
Let y = log(logx)
Then `"dy"/"dx" = "d"/"dx"[log (logx)]`
= `(1)/"logx" . "d"/"dx"(logx)`
= `(1)/"logx" xx (1)/x = (1)/"xlogx"`
and
`(d^2y)/(dx^2) = "d"/"dx"(xlogx)^-1`
= `(-1)(xlogx)^-2."d"/"dx"(xlogx)`
= `(-1)/(xlogx)^2.[x"d"/"dx"(logx) + (logx)."d"/"dx"(x)]`
= `(-1)/(xlogx)^2.[x xx 1/x + (logx) xx 1]`
= `-(1 + logx)/(xlogx)^2`.
APPEARS IN
संबंधित प्रश्न
Differentiate the function with respect to x.
`(x + 1/x)^x + x^((1+1/x))`
Differentiate the function with respect to x.
xsin x + (sin x)cos x
Find `bb(dy/dx)` for the given function:
(cos x)y = (cos y)x
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 x = a (cos t + t sin t) and y = a (sin t – t cos t), find `(d^2y)/dx^2`.
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 `y = sin^-1 x + cos^-1 x , "find" dy/dx`
If ey ( x +1) = 1, then show that `(d^2 y)/(dx^2) = ((dy)/(dx))^2 .`
Find `dy/dx` if y = xx + 5x
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 `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 ey = yx, then show that `"dy"/"dx" = (logy)^2/(log y - 1)`.
If x = 2cos4(t + 3), y = 3sin4(t + 3), show that `"dy"/"dx" = -sqrt((3y)/(2x)`.
If x = log(1 + t2), y = t – tan–1t,show that `"dy"/"dx" = sqrt(e^x - 1)/(2)`.
If x = `(2bt)/(1 + t^2), y = a((1 - t^2)/(1 + t^2)), "show that" "dx"/"dy" = -(b^2y)/(a^2x)`.
Differentiate 3x w.r.t. logx3.
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)
Find the nth derivative of the following : log (2x + 3)
Choose the correct option from the given alternatives :
If xy = yx, then `"dy"/"dx"` = ..........
If y = A cos (log x) + B sin (log x), show that x2y2 + xy1 + y = 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 x7 . y5 = (x + y)12, show that `("d"y)/("d"x) = y/x`
`"d"/"dx" [(cos x)^(log x)]` = ______.
If `"f" ("x") = sqrt (1 + "cos"^2 ("x"^2)), "then the value of f'" (sqrtpi/2)` is ____________.
If y = `(1 + 1/x)^x` then `(2sqrt(y_2(2) + 1/8))/((log 3/2 - 1/3))` is equal to ______.
If y = `x^(x^2)`, then `dy/dx` is equal to ______.
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 ______.
The derivative of x2x w.r.t. x is ______.
Find `dy/dx`, if y = (sin x)tan x – xlog x.
If y = `log(x + sqrt(x^2 + 4))`, show that `dy/dx = 1/sqrt(x^2 + 4)`
Find `dy/dx`, if y = (log x)x.
Evaluate:
`int log x dx`
If \[y=x^x+x^{\frac{1}{x}}\] then \[\frac{\mathrm{d}y}{\mathrm{d}x}\] is equal to
