Question

how to solve : tan^ -1 ( sqrt(1 + x^2 -x)) by changing it into the form of tan^-1 (tan of something)?

Answer 1

let y = that whole equation
tan y = that whole equation (without tan^-1)

Answer 2

yeah i know that but i want to differentiate this function.
by changing it to the form tan^-1(tan of some thing )

Answer 3

using common formulas and the chain rule we have\[y=\tan^{-1}(\sqrt{1+x^2-x})\]\[\frac{dy}{dx}=\frac1{1+(\sqrt{1+x^2-x})^2}\cdot(\frac d{dx}\sqrt{1+x^2-x})\]if that's what you're asking

Answer 4

well what i want is to change the function y to a form like
\[\tan^{-1} (\tan t) \] which equals t so that dy/dx can be written as dt/dx .
it is done by some trigonometric substitutions for x in the equation .