Advertisements
Advertisements
рдкреНрд░рд╢реНрди
Differentiate the function with respect to x.
`(x + 1/x)^x + x^((1+1/x))`
Advertisements
рдЙрддреНрддрд░
Let y = `(x + 1/x)^x + x^((1+1/x))` = u +v
Where u = `(x + 1/x)^x` and v = `x ^((1+1/x))`
Differentiating the above w.r.t. x we get
`dy/dx = (du)/dx + (dv)/dx` .....(i)
Now, u = `(x + 1/x)^x`
Taking log on both sides, we get,
= `logu = x log (x + 1/x)` ......(ii)
Differentiating (ii) w.r.t. x, we get
`1/u (du)/dx = x d/dx log (x + 1/x) + log (x + 1/x)(1)`
= `x/(x + 1/x) (1 - 1/x^2) + log (x + 1/x)`
⇒ `(du)/dx = (x + 1/x)^x [x/(x + 1/x)(1 - 1/x^2) + log (x + 1/x)]` ....(iii)
Also, v = `x^((1 + 1/x))`
Taking log on both sides, we get,
log v = `(1 + 1/x) log x` ....(iv)
Differentiating (iv) w.r.t. x, we get,
`1/v (dv)/dx = (1 + 1/x)d/dx log x + log x d/dx (1 + 1/x)`
= `(1 + 1/x) 1/x + log x (-1/x^2)`
`(dv)/dx = x^((1+1/x)) [(1 + 1/x) 1/x + log x (-1/x^2)]` ....(v)
Substituting the value of (iii) and (v) in (i), we get,
`dy/dx = (x + 1/x)^x [x/(x + 1/x) (1 - 1/x^2) + log (x + 1/x)] + x^((1 + 1/x)) [(1 + 1/x) 1/x + log x (-1/x^2)]`
APPEARS IN
рд╕рдВрдмрдВрдзрд┐рдд рдкреНрд░рд╢реНтАНрди
If `y=log[x+sqrt(x^2+a^2)]` show that `(x^2+a^2)(d^2y)/(dx^2)+xdy/dx=0`
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.
(x + 3)2 . (x + 4)3 . (x + 5)4
Differentiate the function with respect to x.
xsin x + (sin x)cos x
Find `bb(dy/dx)` for the given function:
xy = `e^((x - y))`
If u, v and w are functions of x, then show that `d/dx(u.v.w) = (du)/dx v.w + u. (dv)/dx.w + u.v. (dw)/dx` in two ways-first by repeated application of product rule, second by logarithmic differentiation.
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`
Differentiate
log (1 + x2) w.r.t. tan-1 (x)
Find `(d^2y)/(dx^2)` , if y = log x
If `"x"^(5/3) . "y"^(2/3) = ("x + y")^(7/3)` , the show that `"dy"/"dx" = "y"/"x"`
If log (x + y) = log(xy) + p, where p is a constant, then prove that `"dy"/"dx" = (-y^2)/(x^2)`.
If `log_10((x^3 - y^3)/(x^3 + y^3))` = 2, show that `dy/dx = -(99x^2)/(101y^2)`.
`"If" y = sqrt(logx + sqrt(log x + sqrt(log x + ... ∞))), "then show that" dy/dx = (1)/(x(2y - 1).`
If y = `x^(x^(x^(.^(.^.∞))`, then show that `"dy"/"dx" = y^2/(x(1 - logy).`.
If x = 2cos4(t + 3), y = 3sin4(t + 3), show that `"dy"/"dx" = -sqrt((3y)/(2x)`.
Find the second order derivatives of the following : log(logx)
Find the nth derivative of the following : log (2x + 3)
If f(x) = logx (log x) then f'(e) is ______
If y = `log[4^(2x)((x^2 + 5)/sqrt(2x^3 - 4))^(3/2)]`, find `("d"y)/("d"x)`
If log5 `((x^4 + "y"^4)/(x^4 - "y"^4))` = 2, show that `("dy")/("d"x) = (12x^3)/(13"y"^2)`
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`
lf y = `2^(x^(2^(x^(...∞))))`, then x(1 - y logx logy)`dy/dx` = ______
If xy = ex-y, then `"dy"/"dx"` at x = 1 is ______.
`2^(cos^(2_x)`
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 y = `x^(x^2)`, then `dy/dx` is equal to ______.
If `log_10 ((x^3 - y^3)/(x^3 + y^3))` = 2 then `dy/dx` = ______.
The derivative of log x with respect to `1/x` is ______.
Find `dy/dx`, if y = (log x)x.
Evaluate:
`int log x dx`
