Advertisements
Advertisements
Question
Differentiate the function with respect to x.
(log x)x + xlog x
Advertisements
Solution
Let, y = (log x)x + xlog x
Again, let y = u + v
Differentiating both sides with respect to x,
`(dy)/dx = (du)/dx + (dv)/dx` ....(1)
Now, u = (log x)x
Taking logarithm of both sides,
log u = log (log x)x ...[∵ log mn = n log m]
log u = x log (log x)
Differentiating both sides with respect to x,
`1/u (du)/dx = x d/dx log (log x) + log (log x) d/dx (x)`
= `x * 1/(log x) d/dx (log x) + log (log x) xx 1`
= `x * 1/(log x) 1/x + log (log x)`
= `1/(log x) + log (log x)`
= `u [log (log x) + 1/(log x)]`
∴ `(du)/dx = (log x)^x [log (log x) + 1/log x]`
Also v = xlog x
Taking logarithm of both sides,
log v = log xlog x
= log x log x
= (log x)2
Differentiating both sides with respect to x,
`1/v (dv)/dx = d/dx (log x)^2`
= `2 log x d/dx log x`
= `2 log x xx 1/x`
= `v ((2 log x)/x)`
∴ `(dv)/dx = x^(log x)((2 log x)/x)`
From equation (1),
`(dy)/dx = (du)/dx + (dv)/dx`
`∴ dy/dx = (log x)^x [log (log x) + 1/log x] + x^(log x) ((2 log x)/x)`
APPEARS IN
RELATED QUESTIONS
Differentiate the following function with respect to x: `(log x)^x+x^(logx)`
Differentiate the function with respect to x.
xx − 2sin x
Find `bb(dy/dx)` for the given function:
xy + yx = 1
Find `bb(dy/dx)` for the given function:
yx = xy
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`
Evaluate
`int 1/(16 - 9x^2) dx`
Find `(d^2y)/(dx^2)` , if y = log x
Differentiate : log (1 + x2) w.r.t. cot-1 x.
Find `"dy"/"dx"` if y = xx + 5x
If log (x + y) = log(xy) + p, where p is a constant, then prove that `"dy"/"dx" = (-y^2)/(x^2)`.
If `log_5((x^4 + y^4)/(x^4 - y^4)) = 2, "show that""dy"/"dx" = (12x^3)/(13y^3)`.
If x = `asqrt(secθ - tanθ), y = asqrt(secθ + tanθ), "then show that" "dy"/"dx" = -y/x`.
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 = sin–1(et), y = `sqrt(1 - e^(2t)), "show that" sin x + dy/dx` = 0
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.
If y = log (log 2x), show that xy2 + y1 (1 + xy1) = 0.
Find the nth derivative of the following : log (2x + 3)
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)`
If y = 5x. x5. xx. 55 , find `("d"y)/("d"x)`
The rate at which the metal cools in moving air is proportional to the difference of temperatures between the metal and air. If the air temperature is 290 K and the metal temperature drops from 370 K to 330 K in 1 O min, then the time required to drop the temperature upto 295 K.
If y = `{f(x)}^{phi(x)}`, then `dy/dx` is ______
`d/dx(x^{sinx})` = ______
`"d"/"dx" [(cos x)^(log x)]` = ______.
`log [log(logx^5)]`
`lim_("x" -> -2) sqrt ("x"^2 + 5 - 3)/("x" + 2)` is equal to ____________.
If `"y" = "e"^(1/2log (1 + "tan"^2"x")), "then" "dy"/"dx"` is equal to ____________.
If y = `(1 + 1/x)^x` then `(2sqrt(y_2(2) + 1/8))/((log 3/2 - 1/3))` is equal to ______.
Find `dy/dx`, if y = (sin x)tan x – xlog x.
The derivative of log x with respect to `1/x` is ______.
What is the first step in the standard procedure for \[y=[u(x)]^{v(x)}\]?
Which is the derivative of \[\sqrt{\frac{(x-3)(x^2+4)}{3x^2+4x+5}}\]?
What condition must be ensured for an expression inside logarithm?
