Advertisements
Advertisements
प्रश्न
Find the second order derivatives of the following : xx
Advertisements
उत्तर
y = xx
∴ log y = log xx = x log x
Differentiating both sides w.r.t. x, we get
`(1)/y."dy"/"dx" = "d"/"dx"(xlogx)`
∴ `(1)/y."dy"/"dx" = x."d"/"dx"(logx) + (logx)."d"/"dx"(x)`
= `x/x + (logx) (1)`
= 1 + log x
∴ `"dy"/"dx" = y(1 + logx) = x^x(1 + log x)`
∴ `"d"/"dx"(x^x) = x^x(1 + log x)` ...(1)
∴ `(d^2y)/(dx^2) = "d"/"dx"[x^2(1 + log x)]`
= `x^x."d"/"dx"(1 + log x) + (1 + log x)."d"/"dx"(x^x)`
= `x^x(0 + 1/x) + (1 + logx).x^x(1 + logx)` ...[By (1)]
= xx–1 + xx (1 + log x)2.
APPEARS IN
संबंधित प्रश्न
Solve the following differential equation:
x2 dy + (xy + y2) dx = 0, when x = 1 and y = 1
Find `dy/dx`, if `xsqrt(x) + ysqrt(y) = asqrt(a)`.
Find `"dy"/"dx"` if xey + yex = 1
Find `"dy"/"dx"` if ex+y = cos(x – y)
Find `"dy"/"dx"` if cos (xy) = x + y
Find `"dy"/"dx"` if, y = log(log x)
Find `"dy"/"dx"` if, y = `"e"^(5"x"^2 - 2"x" + 4)`
Find `"dy"/"dx"` if, y = `"a"^((1 + log "x"))`
If y = 2x2 + 22 + a2, then `"dy"/"dx" = ?`
If `"x"^"m"*"y"^"n" = ("x + y")^("m + n")`, then `"dy"/"dx" = "______"/"x"`
State whether the following is True or False:
The derivative of polynomial is polynomial.
Find the rate of change of demand (x) of a commodity with respect to its price (y) if y = 25 + 30x – x2.
Find `"dy"/"dx"`, if y = xx.
If y = sec (tan−1x), then `dy/dx` at x = 1 is ______.
If y = f(u) is a differentiable function of u and u = g(x) is a differentiable function of x such that the composite function y = f[g(x)] is a differentiable function of x, then `("d"y)/("d"x) = ("d"y)/("d"u)*("d"u)/("d"x)`. Hence find `("d"y)/("d"x)` if y = sin2x
If x = f(t) and y = g(t) are differentiable functions of t so that y is a differentiable function of x and `(dx)/(dt)` ≠ 0 then `(dy)/(dx) = ((dy)/(dt))/((dx)/(d"))`.
Hence find `(dy)/(dx)` if x = sin t and y = cost
If y = `1/sqrt(3x^2 - 2x - 1)`, then `("d"y)/("d"x)` = ?
Choose the correct alternative:
If y = `x^(sqrt(x))`, then `("d"y)/("d"x)` = ?
If y = x2, then `("d"^2y)/("d"x^2)` is ______
State whether the following statement is True or False:
If y = ex, then `("d"y)/("d"x)` = ex
y = (6x4 – 5x3 + 2x + 3)6, find `("d"y)/("d"x)`
Solution: Given,
y = (6x4 – 5x3 + 2x + 3)6
Let u = `[6x^4 - 5x^3 + square + 3]`
∴ y = `"u"^square`
∴ `("d"y)/"du"` = 6u6–1
∴ `("d"y)/"du"` = 6( )5
and `"du"/("d"x) = 24x^3 - 15(square) + 2`
By chain rule,
`("d"y)/("d"x) = ("d"y)/square xx square/("d"x)`
∴ `("d"y)/("d"x) = 6(6x^4 - 5x^3 + 2x + 3)^square xx (24x^3 - 15x^2 + square)`
If y = `2/(sqrt(a^2 - b^2))tan^-1[sqrt((a - b)/(a + b)) tan x/2], "then" (d^2y)/dx^2|_{x = pi/2}` = ______
If y = `x/"e"^(1 + x)`, then `("d"y)/("d"x)` = ______.
If y = (sin x2)2, then `("d"y)/("d"x)` is equal to ______.
If ex + ey = ex+y , prove that `("d"y)/("d"x) = -"e"^(y - x)`
If x = a sec3θ and y = a tan3θ, find `("d"y)/("d"x)` at θ = `pi/3`
If y = `sec^-1 ((sqrt(x) + 1)/(sqrt(x + 1))) + sin^-1((sqrt(x) - 1)/(sqrt(x) + 1))`, then `"dy"/"dx"` is equal to ______.
If `sqrt(1 - x^2) + sqrt(1 - y^2) = a(x - y)`, prove that `(dy)/(dx) = sqrt((1 - y^2)/(1 - x^2))`.
Let f(x) = log x + x3 and let g(x) be the inverse of f(x), then |64g"(1)| is equal to ______.
If f(x) = `{{:(x^3 + 1",", x < 0),(x^2 + 1",", x ≥ 0):}`, g(x) = `{{:((x - 1)^(1//3)",", x < 1),((x - 1)^(1//2)",", x ≥ 1):}`, then (gof) (x) is equal to ______.
If `d/dx` [f(x)] = ax+ b and f(0) = 0, then f(x) is equal to ______.
Find `"dy"/"dx"` if, `"y" = "e"^(5"x"^2 - 2"x" + 4)`
Solve the following:
If y = `root5((3x^2 + 8x + 5)^4)`, find `dy/dx`
The differential equation of (x - a)2 + y2 = a2 is ______
If x = Φ(t) is a differentiable function of t, then prove that:
`int f(x)dx = int f[Φ(t)]*Φ^'(t)dt`
Hence, find `int(logx)^n/x dx`.
Find `"dy"/"dx"` if, y = `"e"^(5"x"^2 - 2"x" + 4)`
Solve the following:
If y = `root(5)((3"x"^2 + 8"x" + 5)^4)`, find `"dy"/"dx"`
Find the rate of change of demand (x) of a commodity with respect to its price (y) if y = 12 + 10`x + 25x^2`
Solve the following.
If `y=root(5)((3x^2 + 8x + 5)^4)`, find `dy/dx`
If `y = root{5}{(3x^2 + 8x + 5)^4}, "find" dy/dx`.
If y = `root{5}{(3x^2 + 8x + 5)^4)`, find `(dy)/(dx)`
Solve the following:
If y = `root5((3x^2 + 8x + 5)^4)`, find `dy/(dx)`.
