Question

What is the derivative of $\arctan \dfrac{1}{x}$?

Answer 1

The arctan is defined as the inverse tangent function of x when x is real. When the tangent of y is equal to x then $\tan y=x$ . Then the arctangent of x is equal to the inverse tangent function of x, which is equal to y: $\arctan x={{\tan }^{-1}}x=y$ .

Complete step-by-step answer:
The inverse trigonometric functions are the inverses of the trigonometric functions. Inverse trigonometric functions are widely used in engineering, physics, navigation etc.
For every trigonometric function we have an inverse trigonometric function which works in reverse as the trigonometric functions. These inverse trigonometric functions have the same name but with arc in front. So, the inverse of tan is arctan. When we see arctan x we understand it as the angle whose tangent is x. We can apply trigonometric functions to various angles, large, small etc. With positive or negative values both but in inverse trigonometric functions we have the problem as we have an infinite number of angles on which we have the same tangent. In order to solve this problem, we have the range of inverse functions limited in such a way that the inverse functions are one-one, that is there is only one result value for each input value. Range is the set of all possible outputs.
Now, computing the derivative of $\arctan \dfrac{1}{x}$ we will start with basic inverse trigonometric properties.,
In order to solve it quickly what you need to use is use the trigonometric relations as $\arctan \dfrac{1}{x}$=$\ arccot x$
And $\dfrac{d\ arccot x}{dx}=-\dfrac{1}{1+{{x}^{2}}}$
So, this is the required derivative.

Note: We must know the derivatives of inverse trigonometric substitution otherwise it becomes very complicated. Different derivatives and their relations are used to get the value of different functions easily.