The tan function:
The trigonometric function tan which is the acronym of tangent function is defined as the ratio of the opposite side to the adjacent side of the angle of which the tan is to be found. Alternatively, if we have a point P on the unit circle such that the co-ordinates of the point are (x,y), then tan theta of angle subtended at the centre by the arc AP would be y/x. See picture below:
The inverse of tan function:
Inverse of the tan function is also called the arctan function. We know that the tangent function is a periodic function. So it is not a one to one function. However if we suitably restrict the domain, then the tangent function is a one to one function and then its inverse function i.e. arctan can also be defined.
The arctan function is: arctan (y) = {(x,y) | tan x = y}.
Thus, if tan (pi/4) = 1, then we say that arctan(1) = pi/4.
What is the derivative of arctan?
The derivative of arc tan can be calculated as follows:
Let y = arctan x,
Then, x = tan y
So, dx/dy = sec^2 (y)
So, dx/dy = 1 + tan^2(y)
So, dx/dy = 1+x^2
So, dy/dx = 1/(1+x^2)
Thus the derivative of arctan is 1/(1+x^2)
The derivatives of other arc tan function such as derivative of arctan x 2, derivative of arctan 2 ^x, arctan(x^2) etc, can all be found using the chain rule along with the formula for derivative of arc tan. The derivative of arctan function represents the rate of change of the function y with respect to change in x. In other words it is the slope of the tangent at any point x. The graph of the derivative of arc tan can be shown as follows:
In the above picture, the red curve is the graph of arc tan (x) and the blue curve is the graph of derivative of arc tan (x). Note that at the point x=0, the graph of arc tan (x) changes from concave up to concave down. Therefore that is the point of inflection. We see that at that point the graph of the derivative of arc tan (x) has a maximum. Also the graph of the derivative is always positive, therefore arc tan is an always increasing function.
