D[Sqrt[Cos[4x]],x] - Wolfram|Alpha

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Derivative

d/dx(sqrt(cos(4 x))) = -(2 sin(4 x))/sqrt(cos(4 x))

Plot

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Roots

x = (π n)/4, n element Z

Integer root

x = 0

Properties as a real function

Domain

{x element R : 8 x + π>4 π n and 8 x<4 π n + π and n element Z}

Range

R (all real numbers)

Surjectivity

surjective onto R

Parity

odd

Series expansion at x=0

-8 x - (32 x^3)/3 - (1216 x^5)/15 + O(x^6)
(Taylor series)

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Series expansion at x=-π/8

1/sqrt(x + π/8) - 20/3 (x + π/8)^(3/2) + 8/5 (x + π/8)^(7/2) + O((x + π/8)^(11/2))
(Puiseux series)

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Series expansion at x=π/8

-sqrt(8 x - π)/(sqrt(π - 8 x) sqrt(x - π/8)) + (20 sqrt(8 x - π) (x - π/8)^(3/2))/(3 sqrt(π - 8 x)) - (8 sqrt(8 x - π) (x - π/8)^(7/2))/(5 sqrt(π - 8 x)) + O((x - π/8)^(11/2))
(generalized Puiseux series)

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Indefinite integral

integral-(2 sin(4 x))/sqrt(cos(4 x)) dx = sqrt(cos(4 x)) + constant

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