Advertisements
Advertisements
प्रश्न
Find the derivatives of the following:
y = `x^(logx) + (logx)^x`
Advertisements
उत्तर
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
संबंधित प्रश्न
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:
g(t) = t3 cos t
Find the derivatives of the following functions with respect to corresponding independent variables:
y = `tan x/x`
Find the derivatives of the following functions with respect to corresponding independent variables:
y = `sinx/(1 + cosx)`
Find the derivatives of the following functions with respect to corresponding independent variables:
y = `sinx/x^2`
Find the derivatives of the following functions with respect to corresponding independent variables:
y = cosec x . cot x
Find the derivatives of the following functions with respect to corresponding independent variables:
y = (x2 + 5) log(1 + x) e–3x
Find the derivatives of the following functions with respect to corresponding independent variables:
Draw the function f'(x) if f(x) = 2x2 – 5x + 3
Differentiate the following:
y = e–mx
Differentiate the following:
y = `x"e"^(-x^2)`
Differentiate the following:
y = sin2(cos kx)
Find the derivatives of the following:
y = `x^(cosx)`
Find the derivatives of the following:
`x^2/"a"^2 + y^2/"b"^2` = 1
Find the derivatives of the following:
`sqrt(x^2 + y^2) = tan^-1 (y/x)`
Find the derivatives of the following:
x = a (cos t + t sin t); y = a (sin t – t cos t)
Find the derivatives of the following:
Find the derivative of sin x2 with respect to x2
Find the derivatives of the following:
If y = sin–1x then find y”
Find the derivatives of the following:
If y = `(sin^-1 x)/sqrt(1 - x^2)`, show that (1 – x2)y2 – 3xy1 – y = 0
Choose the correct alternative:
If x = a sin θ and y = b cos θ, then `("d"^2y)/("d"x^2)` is
Choose the correct alternative:
If y = `(1 - x)^2/x^2`, then `("d"y)/("d"x)` is
