A tutorial on how to find the first derivative of \( f(x) = \arcsin(\sin(x)) \) and graph \( f \) and \( f' \) for \( x \in \mathbb{R} \).
Since the domain of \( f \) is \( \mathbb{R} \) and \( \sin(x) \) is periodic, then \( f(x) = \arcsin(\sin(x)) \) is also a periodic function.
As \( x \) increases from \( 0 \) to \( \frac{\pi}{2} \), \( \sin(x) \) increases from \( 0 \) to \( 1 \) and \( \arcsin(\sin(x)) \) increases from \( 0 \) to \( \frac{\pi}{2} \). In fact, for \( x \in [0 , \frac{\pi}{2}] \), \( \arcsin(\sin(x)) = x \). As \( x \) increases from \( \frac{\pi}{2} \) to \( \frac{3\pi}{2} \), \( \sin(x) \) decreases from \( 1 \) to \( -1 \) and \( \arcsin(\sin(x)) \) decreases from \( \frac{\pi}{2} \) to \( -\frac{\pi}{2} \). As \( x \) increases from \( \frac{3\pi}{2} \) to \( 2\pi \), \( \sin(x) \) increases from \( -1 \) to \( 0 \) and \( \arcsin(\sin(x)) \) increases from \( -\frac{\pi}{2} \) to \( 0 \).
Since \( \sin(x) \) has a period of \( 2\pi \), \( \arcsin(\sin(x)) \) also has a period of \( 2\pi \). The graph below shows the graphs of \( \arcsin(\sin(x)) \) and \( \sin(x) \) from \( 0 \) to \( 2\pi \).
The graph below shows the graphs of \( \arcsin(\sin(x)) \) and \( \sin(x) \) over 3 periods.
Domain of \( f \): \( (-\infty , +\infty) \)
Range of \( f \): \( \left[-\frac{\pi}{2} , \frac{\pi}{2}\right] \)
\( f(x) \) is a composite function and the derivative is computed using the chain rule as follows: Let \( u = \sin(x) \).
Hence \( f(x) = \arcsin(u(x)) \).
Apply the chain rule of differentiation:
\[ f'(x) = \frac{du}{dx} \cdot \frac{d(\arcsin(u))}{du} = \cos(x) \cdot \frac{1}{\sqrt{1 - u^2}} \]
Substituting \( u = \sin(x) \): \[ f'(x) = \frac{\cos(x)}{\sqrt{1 - \sin^2(x)}} = \frac{\cos(x)}{|\cos(x)|} \]
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 \( x = \frac{\pi}{2} + k\pi \), where \( k \) is an integer. For these same values of \( x \), \( \arcsin(\sin(x)) \) has either a maximum value equal to \( \frac{\pi}{2} \) or a minimum value equal to \( -\frac{\pi}{2} \).
Note that although \( \arcsin(\sin(x)) \) is continuous for all values of \( x \), its derivative is undefined at certain values of \( x \).
