Advertisements
Advertisements
Question
Differentiate the following function with respect to x: `(log x)^x+x^(logx)`
Advertisements
Solution
`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
RELATED QUESTIONS
if xx+xy+yx=ab, then find `dy/dx`.
Differentiate the function with respect to x.
`(sin x)^x + sin^(-1) sqrtx`
Find `bb(dy/dx)` for the given function:
yx = xy
Find `bb(dy/dx)` for the given function:
(cos x)y = (cos y)x
If `y = sin^-1 x + cos^-1 x , "find" dy/dx`
Evaluate
`int 1/(16 - 9x^2) dx`
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 ey = yx, then show that `"dy"/"dx" = (logy)^2/(log y - 1)`.
If x = esin3t, y = ecos3t, then show that `dy/dx = -(ylogx)/(xlogy)`.
If x = 2cos4(t + 3), y = 3sin4(t + 3), show that `"dy"/"dx" = -sqrt((3y)/(2x)`.
If x = sin–1(et), y = `sqrt(1 - e^(2t)), "show that" sin x + dy/dx` = 0
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.
Find the nth derivative of the following: log (ax + b)
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 = `25^(log_5sin_x) + 16^(log_4cos_x)` then `("d"y)/("d"x)` = ______.
If y = log [cos(x5)] then find `("d"y)/("d"x)`
`d/dx(x^{sinx})` = ______
`"d"/"dx" [(cos x)^(log x)]` = ______.
If `("f"(x))/(log (sec x)) "dx"` = log(log sec x) + c, then f(x) = ______.
`log [log(logx^5)]`
If y = `x^(x^2)`, then `dy/dx` is equal to ______.
The derivative of log x with respect to `1/x` is ______.
Find the derivative of `y = log x + 1/x` with respect to x.
What is the first step in the standard procedure for \[y=[u(x)]^{v(x)}\]?
After taking logarithms in logarithmic differentiation, which rules are used to simplify products, quotients and powers before differentiation?
Which derivative correctly represents differentiating \[\ln y\] carefully?
