Find the derivative of F (x) = sin (x / 2)

Find the derivative of F (x) = sin (x / 2)

It can be understood that if y = x / 2, then the original problem becomes to find the derivative of F (x) = siny. Because y here is a composite function, f (x) = y '(siny)' = 1 / 2 * cos (x / 2)

Given that the function y = y (x) is determined by the equation y = sin (x + y), find the derivative of Y

The derivative of X on both sides of equation y = sin (x + y) is:
y'=cos(x+y)(x+y)'=cos(x+y)(1+y')
The results of the transfer arrangement are as follows:
[1-cos(x+y)]y'=cos(x+y)
Therefore: y '= cos (x + y) / [1-cos (x + y)]

Find the derivative of the function y = f (x) determined by the implicit function equation y = sin (x + y)

y'=cos(x+y)(1+y')
y'=cos(x+y)/(1-cos(x+y))

Function f (x) = sin（ ω x+ φ)+ cos( ω x+ φ) ( ω> 0,| φ|< If the minimum period of π / 2) is π and f (- x) = f (x), then A f (x) monotonically decreases on (0, π / 2) B F (x) monotonically decreases on (π / 4,3 π / 4) C f (x) monotonically increases on (0, π / 2) D f (x) monotonically increases on (π / 4,3 π / 4) Please explain in detail and take steps. Thank you

f(x)=sin ( ω x+ φ)+ cos( ω x+ φ)= √2sin ( ω x+ φ+ π/4）
Period T = 2 π/ ω= π
∴ ω= two
F (- x) = f (x), then f (x) is an even function
Then f (0) = ±√ 2
Namely φ+ π/4=kπ+π/2
Namely φ= kπ+π/4
Combined with known, φ= π/4
Then f (x) = √ 2Sin (2x + π / 2) = √ 2cos2x
‡ monotonically decreasing on (0, π / 2), select a

Minimum positive period of function f (x) = sin ^ 4x + cos ^ 4x

Because (sin ^ 2 x + cos ^ 2 x) ^ 2 = sin ^ 4 x + cos ^ 4 x + 2 sin ^ 2 x cos ^ 2 x, f (x) = (sin ^ 2 x + cos ^ 2 x) ^ 2 - 2 sin ^ 2 x cos ^ 2 XF (x) = 1 - 2 sin ^ 2 x cos ^ 2 x = 1 - 1 / 2 sin ^ 2 (2x) = 1 - 1 / 4 * (1 - cos 4x

Known function f (x) = sin (x)+ θ）+ cos（x- θ） Is an even function, then θ The value is __

∵f（-x）=sin（-x+ θ）+ cos（-x- θ）= sin θ cosx-cos θ sinx+cosxcos θ- sinxsin θ= f（x）=sinxcos θ+ cosxsin θ+ cosxcos θ+ sinxsin θ
∴-cos θ sinx-sinxsin θ= f（x）=sinxcos θ+ sinxsin θ
∴-2sinxcos θ= 2sinxsin θ
∴sinx（sin θ+ cos θ）= 0
∴ θ= kπ-π
4，k∈Z
So the answer is: K π - π
4，k∈Z