Derivative Of Arctan ( x )

Derivative Of Arctan ( x )

Many students ask me "How to find the derivative of arctan x" and "What is the derivative of arctan x" 

"Derivative of arctan x".

And according to that we will explain a proper solution in which we will show you how to exactly find the Derivative of arctan x. 

Even, after that you can also able to find the Derivative of arctan of any function. 

So let's start🙂.

Derivative of arctan x

Let's use our formula for the derivative of an inverse function to find the derivative of the inverse of the tangent function.

Y = tan^-1x  = arctan.

We simplify the equation by taking the tangent of both sides:

Y =  tan^-1x

tan y = tan (tan^-1x)

tany = x

To get an idea what to expect, we start by graphing the tangent function (see Figure 1). 

The function tan(x) is defined for [ -π/2 < x < +π/2 ]

It's graph extends from negative infinity to positive infinity.

If we reflect the graph of tan x across the line y = x . we get the graph of

 y= arctan  x (Figure 2).

Note that function of arctan x is defined for all values of x from minus infinity to plus infinity, and

 lim x→∞ (arctan x)= π/2.

Let y = tan(x)

Recall the definition of tan(x) as sin(x)/cos(x)

d/dx ( y ) = d/dx tan(x)
                  =  d (sin x) / dx(cos x)       
                  
With the help of quotient rule,

Which states that for y = f(x)/g(x),
         = dy/dx
         = [(f'(x)g(x) - f(x)g'(x)) /g²(x) ]

 with f(x) = sin(x) and

           g(x) = cos(x).

So,  Derivative of tan(x) is sec²(x).  

Video that helps to find derivative of the arctan x

Hope after this step by step explanation on how to find derivative of arctan x. And now you are able to understand the Derivative of arctan of any function.