Advertisements
Advertisements
Question
Find the derivatives of the following:
y = `x^(logx) + (logx)^x`
Advertisements
Solution
y = `x^(logx) + (logx)^x`
Let u = xlog x, v = (log x)x
log u = log xlog x
log u = (log x)(log x)
log u = (log x)2
`1/u * ("d"u)/("d"x) = 2 log x xx 1/x`
`("d"u)/("d"x) = 2u (logx)/x` ........(1)
v = (log x)x
log v = log (log x)x
log v = x log (log x)
`1/"v" * ("dv")/("d"x) = x xx 1/logx xx 1/x + log(log x) xx 1`
`("dv")/("d"x) = "v"[1/logx + log(logx)]` ........(2)
y = u + v
`("d"y)/("d"x) = ("d"u)/("d"x) + ("dv")/("d"x)`
`("d"y)/("d"x) = 2u logx/x + "v"[1/logx + log (log x)]`
By equation (1) and (2)
`("d"y)/("d"x) = 2 x^(logx) * logx/x + (logx)^x [1/logx + log(logx)]`
APPEARS IN
RELATED QUESTIONS
Find the derivatives of the following functions with respect to corresponding independent variables:
y = sin x + cos x
Find the derivatives of the following functions with respect to corresponding independent variables:
y = x sin x cos x
Find the derivatives of the following functions with respect to corresponding independent variables:
y = sin x0
Differentiate the following:
y = (x2 + 4x + 6)5
Differentiate the following:
F(x) = (x3 + 4x)7
Differentiate the following:
h(t) = `("t" - 1/"t")^(3/2)`
Differentiate the following:
y = `sqrt(x +sqrt(x)`
Differentiate the following:
y = `sin^-1 ((1 - x^2)/(1 + x^2))`
Find the derivatives of the following:
y = `x^(cosx)`
Find the derivatives of the following:
`sqrt(x) = "e"^((x - y))`
Find the derivatives of the following:
`x^2/"a"^2 + y^2/"b"^2` = 1
Find the derivatives of the following:
tan (x + y) + tan (x – y) = x
Find the derivatives of the following:
x = `(1 - "t"^2)/(1 + "t"^2)`, y = `(2"t")/(1 + "t"^2)`
Find the derivatives of the following:
`tan^-1 ((cos x + sin x)/(cos x - sin x))`
Find the derivatives of the following:
Find the derivative of `sin^-1 ((2x)/(1 + x^2))` with respect to `tan^-1 x`
Find the derivatives of the following:
If x = a(θ + sin θ), y = a(1 – cos θ) then prove that at θ = `pi/2`, yn = `1/"a"`
Find the derivatives of the following:
If sin y = x sin(a + y), the prove that `("d"y)/("d"x) = (sin^2("a" + y))/sin"a"`, a ≠ nπ
Choose the correct alternative:
If x = a sin θ and y = b cos θ, then `("d"^2y)/("d"x^2)` is
