Advertisements
Advertisements
प्रश्न
Differentiate the following function with respect to x: `(log x)^x+x^(logx)`
Advertisements
उत्तर
`Let y=(logx)^x+x^(logx).............(1)`
`Now `
`y=y_1+y_2 ..........................(2)`
Differentiating (2) with respect x, we get
`dy/dx=dy_1/dx+dy_2/dx.........(3)`
Now take log of y1 = (log x)x
`log y_1 = x log (log x)`
Differentiating with respect to x, we get
`1/y_2 dy_2/dx=(2logx) xx 1/x`
`dy_2/dx=y_2((2logx)/x)=x^(logx)((2logx)/x)................(5)`
Adding equation (4) and (5), we get:
`dy/dx=(logx)^x(1/logx+log(logx))+x^(logx)((2logx)/x)`
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.
(log x)cos x
Differentiate the function with respect to x.
xx − 2sin x
Differentiate the function with respect to x.
(log x)x + xlog x
Find the derivative of the function given by f(x) = (1 + x) (1 + x2) (1 + x4) (1 + x8) and hence find f′(1).
Differentiate the function with respect to x:
xx + xa + ax + aa, for some fixed a > 0 and x > 0
If x = a (cos t + t sin t) and y = a (sin t – t cos t), find `(d^2y)/dx^2`.
if `x^m y^n = (x + y)^(m + n)`, prove that `(d^2y)/(dx^2)= 0`
If `log_5((x^4 + y^4)/(x^4 - y^4)) = 2, "show that""dy"/"dx" = (12x^3)/(13y^3)`.
`"If" y = sqrt(logx + sqrt(log x + sqrt(log x + ... ∞))), "then show that" dy/dx = (1)/(x(2y - 1).`
If x = esin3t, y = ecos3t, then show that `dy/dx = -(ylogx)/(xlogy)`.
If x = a cos3t, y = a sin3t, show that `"dy"/"dx" = -(y/x)^(1/3)`.
If x = 2cos4(t + 3), y = 3sin4(t + 3), show that `"dy"/"dx" = -sqrt((3y)/(2x)`.
If x = `(2bt)/(1 + t^2), y = a((1 - t^2)/(1 + t^2)), "show that" "dx"/"dy" = -(b^2y)/(a^2x)`.
Find the second order derivatives of the following : log(logx)
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)
If y = `25^(log_5sin_x) + 16^(log_4cos_x)` then `("d"y)/("d"x)` = ______.
If y = `log[sqrt((1 - cos((3x)/2))/(1 +cos((3x)/2)))]`, find `("d"y)/("d"x)`
If y = `(sin x)^sin x` , then `"dy"/"dx"` = ?
Derivative of loge2 (logx) with respect to x is _______.
If `("f"(x))/(log (sec x)) "dx"` = log(log sec x) + c, then f(x) = ______.
Derivative of `log_6`x with respect 6x to is ______
If xm . yn = (x + y)m+n, prove that `"dy"/"dx" = y/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 ______.
The derivative of log x with respect to `1/x` is ______.
