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)`
if xx+xy+yx=ab, then find `dy/dx`.
Differentiate the function with respect to x.
xx − 2sin 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 x = a (cos t + t sin t) and y = a (sin t – t cos t), find `(d^2y)/dx^2`.
If ey ( x +1) = 1, then show that `(d^2 y)/(dx^2) = ((dy)/(dx))^2 .`
Find `(dy)/(dx) , if y = sin ^(-1) [2^(x +1 )/(1+4^x)]`
If y = (log x)x + xlog x, find `"dy"/"dx".`
If xy = ex–y, then show that `"dy"/"dx" = logx/(1 + logx)^2`.
If y = `x^(x^(x^(.^(.^.∞))`, then show that `"dy"/"dx" = y^2/(x(1 - logy).`.
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)`.
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.
If y = log (log 2x), show that xy2 + y1 (1 + xy1) = 0.
Find the nth derivative of the following : log (2x + 3)
If y = A cos (log x) + B sin (log x), show that x2y2 + xy1 + y = 0.
If y = log [cos(x5)] then find `("d"y)/("d"x)`
If y = `log[sqrt((1 - cos((3x)/2))/(1 +cos((3x)/2)))]`, find `("d"y)/("d"x)`
If xy = ex-y, then `"dy"/"dx"` at x = 1 is ______.
If y = tan-1 `((1 - cos 3x)/(sin 3x))`, then `"dy"/"dx"` = ______.
If `("f"(x))/(log (sec x)) "dx"` = log(log sec x) + c, then f(x) = ______.
If y = `("e"^"2x" sin x)/(x cos x), "then" "dy"/"dx" = ?`
Derivative of `log_6`x with respect 6x to is ______
`8^x/x^8`
`log (x + sqrt(x^2 + "a"))`
`log [log(logx^5)]`
If xm . yn = (x + y)m+n, prove that `"dy"/"dx" = y/x`
`lim_("x" -> 0)(1 - "cos x")/"x"^2` is equal to ____________.
If `"f" ("x") = sqrt (1 + "cos"^2 ("x"^2)), "then the value of f'" (sqrtpi/2)` is ____________.
If `"y" = "e"^(1/2log (1 + "tan"^2"x")), "then" "dy"/"dx"` is equal to ____________.
If y `= "e"^(3"x" + 7), "then the value" |("dy")/("dx")|_("x" = 0)` is ____________.
If `f(x) = log [e^x ((3 - x)/(3 + x))^(1/3)]`, then `f^'(1)` is equal to
If `log_10 ((x^2 - y^2)/(x^2 + y^2))` = 2, then `dy/dx` is equal to ______.
