Advertisements
Advertisements
Question
Differentiate the function with respect to x.
`(x cos x)^x + (x sin x)^(1/x)`
Advertisements
Solution
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)` ...[∵ 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 of 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
RELATED QUESTIONS
Differentiate the function with respect to x.
xsin x + (sin 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 = `e^((x - y))`
Differentiate the function with respect to x:
xx + xa + ax + aa, for some fixed a > 0 and x > 0
If ey ( x +1) = 1, then show that `(d^2 y)/(dx^2) = ((dy)/(dx))^2 .`
Evaluate
`int 1/(16 - 9x^2) dx`
Find `dy/dx` if y = xx + 5x
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 `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 = `asqrt(secθ - tanθ), y = asqrt(secθ + tanθ), "then show that" "dy"/"dx" = -y/x`.
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
Differentiate 3x w.r.t. logx3.
If y = log (log 2x), show that xy2 + y1 (1 + xy1) = 0.
Find the nth derivative of the following: log (ax + b)
Choose the correct option from the given alternatives :
If xy = yx, then `"dy"/"dx"` = ..........
If y = A cos (log x) + B sin (log x), show that x2y2 + xy1 + y = 0.
If y = `25^(log_5sin_x) + 16^(log_4cos_x)` then `("d"y)/("d"x)` = ______.
If y = `log[sqrt((1 - cos((3x)/2))/(1 +cos((3x)/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.
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 ______.
If `("f"(x))/(log (sec x)) "dx"` = log(log sec x) + c, then f(x) = ______.
`2^(cos^(2_x)`
`log (x + sqrt(x^2 + "a"))`
If xm . yn = (x + y)m+n, prove that `"dy"/"dx" = y/x`
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.
Evaluate:
`int log x dx`
