# 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.

## Graphs of tan(x) and arctan(tan(x))

f(x) is defined for all values x in R except x = ?/2 + k * ?, 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 -?/2 to ?/2 exclusive, tan(x) increases from infinitely small values (- infinity) to infinitely large values (+ infinity) and arctan(tan(x)) increases from -?/2 to ?/2 exclusive since tan(x) is undefined at -?/2 and + ?/2. In fact for x in (-?/2 , ?/2) arctan(tan(x)) = x.

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

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

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

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

## Derivative of f(x) = arctan(tan(x)) and its Derivative

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 / (u^{2}) + 1)

= (1 / cos(x)^{2}) * 1 / (tan(x)^{2}(x) + 1)

= 1 , for x not equal to ?/2 + k*?/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 = ?/2 + k * ?, where k is an integer. Note that f(x) is undefined for these same values of x.

### More References and links

Graph, Domain and Range of Arctan function

differentiation and derivatives