Advertisements
Advertisements
प्रश्न
Differentiate xx w.r.t. xsix.
उत्तर
Let u = xx and v = xsinx
Then we want to find `"du"/"dx"`.
Take, u = xx
∴ log u = x log x
Differentiating both sides w.r.t. x, we get
`(1)/u."du"/"dx" = "d"/"dx"(xlogx)`
= `x"d"/"dx"(logx) + (logx)."d"/"dx"(x)`
= `x xx (1)/x + (logx) xx 1`
∴ `"du"/"dx" = u(1 + logx)`
= xx(1 + log x)
Also, v = xsinx
∴ log v = logxsinx = (sin x)(log x)
Differentiating both sides w.r.t. x, we get
`(1)/v."dv"/"dx" = "d"/"dx"[(sin x)(logx)]`
= `(sinx)."d"/"dx"(logx) + (logx)."d"/"dx"(sinx)`
= `(sinx) xx (1)/x + (logx)(cosx)`
∴ `"dv"/"dx" = v[sinx/x + (logx)(cosx)]`
= `x^(sinx)[sinx/x + (logx)(cosx)]`
∴ `"du"/"dv" = (("du"/"dx"))/(("dv"/"dx")`
= `(x^x(1 + logx))/(x^(sinx)[sinx/x + (logx)(cosx)]`
= `(x^x(1 + log x) xx x)/(x^(sinx)[sinx + x cosx.logx]`
= `((1 + logx).x^(x+ 1 - sinx))/(sinx + xcosx.logx)`.
APPEARS IN
संबंधित प्रश्न
Find `dx/dy` in the following.
x2 + xy + y2 = 100
if `(x^2 + y^2)^2 = xy` find `(dy)/(dx)`
Show that the derivative of the function f given by
If for the function
\[\Phi \left( x \right) = \lambda x^2 + 7x - 4, \Phi'\left( 5 \right) = 97, \text { find } \lambda .\]
Examine the differentialibilty of the function f defined by
\[f\left( x \right) = \begin{cases}2x + 3 & \text { if }- 3 \leq x \leq - 2 \\ \begin{array}xx + 1 \\ x + 2\end{array} & \begin{array} i\text { if } - 2 \leq x < 0 \\\text { if } 0 \leq x \leq 1\end{array}\end{cases}\]
If f (x) = |x − 2| write whether f' (2) exists or not.
Let \[f\left( x \right)\begin{cases}a x^2 + 1, & x > 1 \\ x + 1/2, & x \leq 1\end{cases}\] . Then, f (x) is derivable at x = 1, if
Find `"dy"/"dx"` ; if x = sin3θ , y = cos3θ
Find `(dy)/(dx) if y = cos^-1 (√x)`
Discuss extreme values of the function f(x) = x.logx
If y = `sqrt(cosx + sqrt(cosx + sqrt(cosx + ... ∞)`, then show that `"dy"/"dx" = sinx/(1 - 2y)`.
Find `"dy"/"dx"`, if : x = `sqrt(a^2 + m^2), y = log(a^2 + m^2)`
Find `"dy"/"dx"`, if : x = sinθ, y = tanθ
Find `"dy"/"dx"`, if : x = a(1 – cosθ), y = b(θ – sinθ)
Find `"dy"/"dx"`, if : `x = cos^-1(4t^3 - 3t), y = tan^-1(sqrt(1 - t^2)/t)`.
Find `"dy"/"dx"` if : x = a cos3θ, y = a sin3θ at θ = `pi/(3)`
Find `"dy"/"dx"` if : x = t2 + t + 1, y = `sin((pit)/2) + cos((pit)/2) "at" t = 1`
Find `dy/dx` if : x = 2 cos t + cos 2t, y = 2 sin t – sin 2t at t = `pi/(4)`
If x = `(t + 1)/(t - 1), y = (t - 1)/(t + 1), "then show that" y^2 + "dy"/"dx"` = 0.
DIfferentiate x sin x w.r.t. tan x.
Find `(d^2y)/(dx^2)` of the following : x = a(θ – sin θ), y = a(1 – cos θ)
Find `(d^2y)/(dx^2)` of the following : x = sinθ, y = sin3θ at θ = `pi/(2)`
If x = at2 and y = 2at, then show that `xy(d^2y)/(dx^2) + a` = 0.
If y = `e^(mtan^-1x)`, show that `(1 + x^2)(d^2y)/(dx^2) + (2x - m)"dy"/"dx"` = 0.
If y = eax.sin(bx), show that y2 – 2ay1 + (a2 + b2)y = 0.
Find the nth derivative of the following : (ax + b)m
Find the nth derivative of the following : apx+q
Find the nth derivative of the following:
y = e8x . cos (6x + 7)
Choose the correct option from the given alternatives :
If y = sec (tan –1x), then `"dy"/"dx"` at x = 1, is equal to
Choose the correct option from the given alternatives :
If y = sin (2sin–1 x), then dx = ........
Choose the correct option from the given alternatives :
If `xsqrt(y + 1) + ysqrt(x + 1) = 0 and x ≠ y, "then" "dy"/"dx"` = ........
If y `tan^-1(sqrt((a - x)/(a + x)))`, where – a < x < a, then `"dy"/"dx"` = .........
Choose the correct option from the given alternatives :
If y = `a cos (logx) and "A"(d^2y)/(dx^2) + "B""dy"/"dx" + "C"` = 0, then the values of A, B, C are
Solve the following :
f(x) = –x, for – 2 ≤ x < 0
= 2x, for 0 ≤ x < 2
= `(18 - x)/(4)`, for 2 < x ≤ 7
g(x) = 6 – 3x, for 0 ≤ x < 2
= `(2x - 4)/(3)`, for 2 < x ≤ 7
Let u (x) = f[g(x)], v(x) = g[f(x)] and w(x) = g[g(x)]. Find each derivative at x = 1, if it exists i.e. find u'(1), v' (1) and w'(1). If it doesn't exist, then explain why?
If sin y = x sin (a + y), then show that `"dy"/"dx" = (sin^2(a + y))/(sina)`.
Differentiate log `[(sqrt(1 + x^2) + x)/(sqrt(1 + x^2 - x)]]` w.r.t. cos (log x).
If log y = log (sin x) – x2, show that `(d^2y)/(dx^2) + 4x "dy"/"dx" + (4x^2 + 3)y` = 0.
If x= a cos θ, y = b sin θ, show that `a^2[y(d^2y)/(dx^2) + (dy/dx)^2] + b^2` = 0.
Find `"dy"/"dx"` if, x3 + x2y + xy2 + y3 = 81
Solve the following:
If `"x"^5 * "y"^7 = ("x + y")^12` then show that, `"dy"/"dx" = "y"/"x"`
Choose the correct alternative.
If y = 5x . x5, then `"dy"/"dx" = ?`
State whether the following is True or False:
The derivative of `"x"^"m"*"y"^"n" = ("x + y")^("m + n")` is `"x"/"y"`
If `"x"^7*"y"^9 = ("x + y")^16`, then show that `"dy"/"dx" = "y"/"x"`
If `"x"^"a"*"y"^"b" = ("x + y")^("a + b")`, then show that `"dy"/"dx" = "y"/"x"`
If x2 + y2 = 1, then `(d^2x)/(dy^2)` = ______.
If y = `sqrt(tansqrt(x)`, find `("d"y)/("d"x)`.
If x = sin θ, y = tan θ, then find `("d"y)/("d"x)`.
Differentiate w.r.t x (over no. 24 and 25) `e^x/sin x`
y = `e^(x3)`
Find `(dy)/(dx)` if x + sin(x + y) = y – cos(x – y)
Find `(d^2y)/(dy^2)`, if y = e4x
If y = y(x) is an implicit function of x such that loge(x + y) = 4xy, then `(d^2y)/(dx^2)` at x = 0 is equal to ______.
If 2x + 2y = 2x+y, then `(dy)/(dx)` is equal to ______.
If y = `sqrt(tan x + sqrt(tanx + sqrt(tanx + .... + ∞)`, then show that `dy/dx = (sec^2x)/(2y - 1)`.
Find `dy/dx` at x = 0.
Find `dy/dx if, x= e^(3t), y = e^sqrtt`
If log(x+y) = log(xy) + a then show that, `dy/dx = (-y^2)/x^2`
If log (x + y) = log (xy) + a then show that, `dy/dx = (−y^2)/x^ 2`
Find `dy/dx if , x = e^(3t) , y = e^sqrtt`
If log(x + y) = log(xy) + a then show that, `dy/dx = (-y^2)/x^2`
Find `dy/dx"if", x= e^(3t), y=e^sqrtt`
Find `dy/dx` if, `x = e^(3t), y = e^(sqrtt)`
