Advertisements
Advertisements
рдкреНрд░рд╢реНрди
Differentiate the function with respect to x.
`(x cos x)^x + (x sin x)^(1/x)`
Advertisements
рдЙрддреНрддрд░
Let, y = `(x cos x)^x + (x sin x)^(1/x)`
Differentiating both sides with respect to x,
`dy/dx = (du)/dx + (dv)/dx` ..(1)
Now, u = (x cos x)x
Taking logarithm of both sides,
log u = log (x cos x)x
log u = x log (x cos x)
Differentiating both sides with respect to x,
`1/u (du)/dx = x d/dx log (x cos x) + log (x cos x) d/dx (x)`
= `x * 1/(x cos x) d/dx (x cos x) + log (x cos x) xx 1`
= `1/(cos x) [x d/dx cos x + cos x d/dx (x)] + log (x cos x)`
= `1/(cos x) [x (- sin x) + cos x xx (1)] + log (x cos x)` ...[тИ╡ loge mn = loge m + loge n]
= `- x (sin x)/(cos x) + (cos x)/(cos x) + log x + log cos x`
= −x tan x + 1 + log x + log cos x
`therefore (du)/dx` = u [1 − x tan x + log x + log cos x]
= (x cos x)x [1 − x tan x + log x + log cos x] ...(2)
Also, v = `(x sin x)^(1/x)`
Taking logarithm of both sides,
log v = `log (x sin x)^(1/x)`
log v = `1/x log (x sin x)`
Differentiating both sides with respect to x,
`1/v (dv)/dx = 1/x d/dx log (x sin x) + log (x sin x) d/dx 1/x`
= `1/x 1/(x sin x) * d/dx (x sin x) + log (x sin x) (-1) x^-2`
= `1/(x^2 sin x) [x d/dx sin x + sin x d/dx (x)] + (log x + log sin x)(-1) x^-2`
= `1/(x^2 sin x)` [x cos x + sin x] `- 1/x^2 log x - 1/x^2 log sin x`
= `1/x^2` [1 + x cot x − log (x sin x)]
`therefore (dv)/dx = v * 1/x^2` [1 + x cot x − log (x sin x)]
= `(x sin x)^(1/x) * 1/x^2` [1 + x cot x − log (x sin x)] ...(3)
Putting the values тАЛтАЛof from equation (2) and (3) in equation (1),
∴ `dy/dx = (du)/dx + (dv)/dx`
= `(x cos x)^x [1 − x tan x + log x + log cos x] + (x sin x)^(1/x) * 1/x^2 [1 + x cot x - log (x sin x)]`
APPEARS IN
рд╕рдВрдмрдВрдзрд┐рдд рдкреНрд░рд╢реНтАНрди
Differentiate the following function with respect to x: `(log x)^x+x^(logx)`
Differentiate the function with respect to x.
(x + 3)2 . (x + 4)3 . (x + 5)4
Find `bb(dy/dx)` for the given function:
(cos x)y = (cos y)x
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 x = a (cos t + t sin t) and y = a (sin t – t cos t), find `(d^2y)/dx^2`.
if `x^m y^n = (x + y)^(m + n)`, prove that `(d^2y)/(dx^2)= 0`
If ey ( x +1) = 1, then show that `(d^2 y)/(dx^2) = ((dy)/(dx))^2 .`
Find `dy/dx` if y = xx + 5x
Differentiate
log (1 + x2) w.r.t. tan-1 (x)
Find `(d^2y)/(dx^2)` , if y = log x
Differentiate : log (1 + x2) w.r.t. cot-1 x.
Solve the following differential equation: (3xy + y2) dx + (x2 + xy) dy = 0
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 ey = yx, then show that `"dy"/"dx" = (logy)^2/(log y - 1)`.
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
If y = `log(x + sqrt(x^2 + a^2))^m`, show that `(x^2 + a^2)(d^2y)/(dx^2) + x "d"/"dx"` = 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 = `log[4^(2x)((x^2 + 5)/sqrt(2x^3 - 4))^(3/2)]`, 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 xy = ex-y, then `"dy"/"dx"` at x = 1 is ______.
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" = ?`
`8^x/x^8`
If y = `log ((1 - x^2)/(1 + x^2))`, then `"dy"/"dx"` is equal to ______.
If `"y" = "e"^(1/2log (1 + "tan"^2"x")), "then" "dy"/"dx"` is equal to ____________.
If `log_10 ((x^3 - y^3)/(x^3 + y^3))` = 2 then `dy/dx` = ______.
Find `dy/dx`, if y = (sin x)tan x – xlog x.
If y = `log(x + sqrt(x^2 + 4))`, show that `dy/dx = 1/sqrt(x^2 + 4)`
If y = `9^(log_3x)`, find `dy/dx`.
Find the derivative of `y = log x + 1/x` with respect to x.
If \[y=x^x+x^{\frac{1}{x}}\] then \[\frac{\mathrm{d}y}{\mathrm{d}x}\] is equal to
If \[y=[u(x)]^{v(x)}\], which expression gives \[\frac{dy}{dx}\] in logarithmic differentiation?
For \[y=x^{\sin x}\], \[x>0\], what equation results after taking logarithm on both sides?
What is \[\frac{dy}{dx}\] for \[y=x^{\sin x}\], \[x>0\]?
What condition must be ensured for an expression inside logarithm?
