Find Derivative of f(x) = arctan(tan(x)) and graph it

A calculus tutorial on how to find the first derivative of f(x) = arctan(tan(x)) and graph f and f ' in its natural domain.

f(x) is defined for all values x in R except x = pi/2 + k * Pi, where k is an integer. Since tan(x) is periodic, then f(x) = arctan(tan(x)) is also a periodic function.

As x increases from -pi/2 to Pi/2 exclusive, tan(x) increases from infinitely small values (- infinity) to infinitely large values (+ infinity) and arctan(tan(x)) increases from -pi/2 to Pi/2 exclusive since tan(x) is undefined at -pi/2 and + pi/2. In fact for x in (-pi/2 , pi/2) arctan(tan(x)) = x.

Since tan(x) has a period of Pi, arctan(tan(x)) also has a period of Pi. The graph below shows the graphs of arctan(tan(x)) and tan(x) from -pi/2 to 3Pi/2.

Domain of f: R - {pi + k*pi/2 , k integer}

Range of f: (-pi/2 , pi/2)

The graph below shows the graphs of arctan(tan(x)).

Graph of tan(x) and arctan(tan(x)) over 3 periods

Derivative of f(x) = arctan(tan(x))

f(x) is a composite function and the derivative is computed using the
chain ruleas follows: Let u = tan(x)

Hence f(x) = arctan(u(x))

Apply the chain rule of differentiation

f '(x) = du/dx d(arctan(u))/du = (1 / cos(x)
2) * 1 / (u2) + 1)

= (1 / cos(x)
2) * 1 / (tan(x)2(x) + 1)

= 1 , for x not equal to pi/2 + k*pi/2 where k is an integer.

Below is shown arctan(tan(x)) in red and its derivative in blue. Note that the derivative is undefined for values of x for which cos(x) is equal to 0, which means at x = pi/2 + k * pi, where k is an integer. Note that f(x) is undefined for these same values of x.

Graph of derivative of arctan(tan(x))

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differentiation and derivatives