Advertisements
Advertisements
प्रश्न
Differentiate the function with respect to x.
(log x)x + xlog x
Advertisements
उत्तर
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
संबंधित प्रश्न
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.
`x^(xcosx) + (x^2 + 1)/(x^2 -1)`
Find `bb(dy/dx)` for the given function:
xy + yx = 1
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 `x^m y^n = (x + y)^(m + n)`, prove that `(d^2y)/(dx^2)= 0`
Evaluate
`int 1/(16 - 9x^2) dx`
Differentiate
log (1 + x2) w.r.t. tan-1 (x)
Find `"dy"/"dx"` if y = xx + 5x
If `log_10((x^3 - y^3)/(x^3 + y^3))` = 2, show that `dy/dx = -(99x^2)/(101y^2)`.
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 = `asqrt(secθ - tanθ), y = asqrt(secθ + tanθ), "then show that" "dy"/"dx" = -y/x`.
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)`.
Find the second order derivatives of the following : x3.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.
Choose the correct option from the given alternatives :
If xy = yx, then `"dy"/"dx"` = ..........
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`
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 y = `log ((1 - x^2)/(1 + x^2))`, then `"dy"/"dx"` is equal to ______.
`lim_("x" -> 0)(1 - "cos x")/"x"^2` is equal to ____________.
If y `= "e"^(3"x" + 7), "then the value" |("dy")/("dx")|_("x" = 0)` is ____________.
If y = `x^(x^2)`, then `dy/dx` is equal to ______.
Find `dy/dx`, if y = (log x)x.
Evaluate:
`int log x dx`
For which type of function is logarithmic differentiation especially useful?
What is \[\frac{1}{y}\cdot\frac{dy}{dx}\] for \[y=\sqrt{\frac{(x-3)(x^2+4)}{3x^2+4x+5}}\]?
What is \[\frac{dy}{dx}\] for \[y=x^{\sin x}\], \[x>0\]?
What condition must be ensured for an expression inside logarithm?
