Find Derivative of f(x) = arcsin(sin(x)) and graph it
A calculus tutorial on how to find the first derivative of f(x) = arcsin(sin(x)) and graph f and f' for x in R.
Since the domain of f is R and sin(x) is periodic, then f(x) = arcsin(sin(x)) is also a periodic function.
As x increases from 0 to Pi/2, sin(x) increases from 0 to 1 and arcsin(sin(x)) increases from 0 to Pi/2. In fact for x in [0 , pi/2] arcsin(sin(x)) = x. As x increases from [Pi/2 , 3Pi/2], sin(x) decreases from 1 to 1 and arcsin(sin(x)) decreases from Pi/2 to Pi/2. As x increases from 3Pi/2 to 2Pi, sin(x) increases from 1 to 0 and arcsin(sin(x)) increases from 3pi/2 to 2Pi.
Since sin(x) has a period of 2Pi, arcsin(sin(x)) also has a period of 2Pi. The graph below shows the graphs of arcsin(sin(x)) and sin(x) from 0 to 2Pi.
The graph below shows the graphs of arcsin(sin(x)) and sin(x) over 3 periods.
Domain of f: (infinity , +infinity)
Range of f: [pi/2 , pi/2]
Derivative of f(x) = arcsin(sin(x))
f(x) is a composite function and the derivative is computed using the chain rule as follows: Let u = sin(x)
Hence f(x) = arctan(u(x))
Apply the chain rule of differentation
f '(x) = du/dx d(arcsin(u))/du = cos(x) * 1 / sqrt(1  u^{2})
= cos(x) * 1 / (1  sin^{2}(x))
= cos(x) / sqrt(sin^{2}(x))
= cos(x) /  cos(x) 
Below is shown arcsin(sin(x)) in red and its derivative in blue. Note that the derivative is undefined for values of x for which cos(x) = 0, which means at x = pi/2 + k*pi, where k is an integer. For these same values of x, arcsin(sin(x)) has either a maximum value equal to pi/2 or a minimum value equal to pi/2.
Note that although arcsin(sin(x)) is continuous for all values of x its derivative is undefined at certain values of x.
More on differentiation and derivatives 

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Updated: 2 April 2013
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