find the slope of the line tangent to the graph of ( y = \tan^{-1}x ) at ( x = 2 ).\nthe slope of the…

find the slope of the line tangent to the graph of ( y = \tan^{-1}x ) at ( x = 2 ).\nthe slope of the tangent line is \n(type an integer or a simplified fraction.)

find the slope of the line tangent to the graph of ( y = \tan^{-1}x ) at ( x = 2 ).\nthe slope of the tangent line is \n(type an integer or a simplified fraction.)

Answer

Explanation:

Step1: Recall the derivative formula

The derivative of (y = \tan^{- 1}x) is (y^\prime=\frac{1}{1 + x^{2}}) (by the formula for the derivative of the inverse - tangent function ((\tan^{-1}u)^\prime=\frac{u^\prime}{1 + u^{2}}), here (u = x) and (u^\prime=1)).

Step2: Substitute (x = 2) into the derivative

When (x = 2), we substitute (x) into (y^\prime=\frac{1}{1 + x^{2}}). So (y^\prime|{x = 2}=\frac{1}{1+(2)^{2}}). Calculate (1+(2)^{2}=1 + 4=5). Then (y^\prime|{x = 2}=\frac{1}{5}).

Answer:

(\frac{1}{5})