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.

Graph of sin(x) and arcsin(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

= cos(x) * 1 / (1 - sin

= cos(x) / sqrt(sin

= 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.

Graph of derivative of arcsin(sin(x))

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differentiation and derivatives