Question

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

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:

\(\frac{1}{5}\)